1. A Numerical Solution to the Flow Near an Infinite Rotating Disk White, Section 3-8.2 MAE 5130: Viscous Flows December 12, 2006 Adam Linsenbardt

2. Flow Near an Infinite Rotating Disk Infinite plane (z=0) rotates at constant angular velocity, ω, about the r-axis Fluid near the wall picks up the circumferential motion Pressure is a function of z only and can’t balance the centripetal acceleration (v r > 0) Conservation of mass requires an axial flow toward the disk to balance the outward radial flow Operating principle behind von Karman’s viscous pump OBJECTIVE: To develop expressions for v r, v θ, v z, and p as well as other useful quantities for studying this type of flow and its applications

3. Setting Up the Problem Conservation of mass and momentum in cylindrical coordinates for the case of the infinite rotating flat plate: Continuity: r: θ:θ: z:

4. Setting Up the Problem Boundary conditions for the conservation of mass and momentum equations: At As

5. Finding the Solution Substituting the following equations with the non-dimensional parameters (F, G, H, P) into the conservation of mass and momentum equations yields the four coupled differential equations to the right which are only functions of z:

6. Finding the Solution Transforming the boundary conditions to be used with the modified differential equations yields: At As

7. Runge-Kutta Solution At As F’(0)= 0.5102 G’(0)= -0.6159

8. Solution Results Numerical Solution for the Rotating Disk z*FF'GG'H-P 0.00.00000.51021.0000-0.61590.0000 0.20.08360.33380.8780-0.5987-0.01790.1674 0.40.13640.19990.7621-0.5577-0.06280.2747 0.60.16600.10150.6557-0.5048-0.12390.3396 0.80.17890.03170.5605-0.4476-0.19330.3765 1.00.1802-0.01570.4766-0.3911-0.26550.3957 1.20.1738-0.04610.4038-0.3381-0.33640.4042 1.40.1626-0.06400.3411-0.2898-0.40380.4068 1.60.1487-0.07280.2875-0.2469-0.46620.4064 1.80.1339-0.07540.2419-0.2095-0.52260.4045 2.00.1189-0.07390.2034-0.1771-0.57320.4022 2.20.1044-0.06980.1708-0.1493-0.61790.4000 2.40.0910-0.06430.1434-0.1257-0.65690.3981 2.60.0788-0.05800.1203-0.1057-0.69080.3965 2.80.0678-0.05170.1009-0.0888-0.72010.3952 3.00.0581-0.04550.0846-0.0745-0.74530.3942 3.20.0496-0.03970.0709-0.0625-0.76680.3934 3.40.0422-0.03440.0594-0.0524-0.78510.3929 3.60.0358-0.02960.0498-0.0439-0.80070.3925 3.80.0303-0.02530.0418-0.0368-0.81390.3921 4.00.0257-0.02160.0350-0.0309-0.82510.3919 4.50.0168-0.01440.0226-0.0198-0.84600.3916 5.00.0109-0.00940.0145-0.0128-0.85960.3915 5.50.0070-0.00620.0094-0.0082-0.86840.3914 6.00.0045-0.00400.0061-0.0053-0.87410.3914 7.00.0019-0.00170.0026-0.0022-0.88010.3913 8.00.0008-0.00070.0011-0.0009-0.88260.3913 9.00.0003-0.00030.0005-0.0004-0.88370.3913 10.00.0001-0.00010.0003-0.0002-0.88410.3913 Compare to Table 3-5, Figure 3-29 in White. Non-dimensional parameters: F = u r * G = u θ * H = u z * -P = -p*

9. Circumferential Velocity Distribution

10. Radial Velocity Distribution

11. Axial Velocity Distribution

12. Other Important Expressions The displacement boundary layer thickness from the plate can be defined by the point where the circumferential velocity of the plate reaches 1% of the wall velocity. From the table, G = 0.01 around z* = 5.4, so: The circumferential wall shear stress on the disk is useful for determining the required torque to operate the rotating disk with a given fluid and angular velocity

13. Other Important Expressions An expression for the torque coefficient for a two-sided disk is then: This expression shows the same dependence on the square root of the Reynolds number as in other expressions derived throughout the course

14. Instability and Turbulence A photograph of laminar spiral vortices and turbulent behavior over a 40 cm diameter disk rotating at 1800 rpm in air. From Figure 3-31 (White) Re ~ 8.8*10 4 R ~ 12 cm Re ~ 3.2*10 5 R ~ 16 cm

15. Conclusions Pump performance increases with increases in viscosity and angular velocity Can easily solve for axial, circumferential, and radial components of velocity using the table and relations Similarity variables and Runge-Kutta method applied to true 3D flows Operating principle of viscous drag pumps (micro-scale, etc.) Extension to other interesting rotating problems (rotating flow next to a fixed disk (tea leaves, secondary flows), rotation of upper stage rocket propellant tanks)