2. Content  Classical Types of Taylor Vortices  Important Parameters  Mesh Independence Study  Taylor-Couette Validation  Wavy Taylor Validation  Turbulent Validation  Thermal Validation  Simple Model Test  Plans for Next Period

3. Classical Types of Taylor VorticesClassical Types of Taylor Vortices Laminar Couette Flow According to the stability analysis without considering the viscous, the flow inside the cylinders should be always instable when outer cylinder is kept fixed. However, viscosity has an important stabilizing influence at low Reynolds numbers. Stability will be broken only if the angular velocity of inner cylinder exceeds a critical value.

4. Classical Types of Taylor VorticesClassical Types of Taylor Vortices Taylor vortex forms when Re exceeds Re c When the angular velocity of the inner cylinder is increased above a certain threshold, Couette laminar flow becomes unstable and a secondary steady state characterized by axis-symmetric toroidal vortices, known as Taylor vortex flow, emerges.

5. Classical Types of Taylor VorticesClassical Types of Taylor Vortices Wavy taylor vortex forms when Re exceeds Re c2 Subsequently increasing the angular speed of the cylinder the system undergoes a progression of instabilities which lead to states with greater spatio-temporal complexity, with the next state being called as Wavy Vortex Flow. The rotational speed approximately 20% higher than the critical speed for transition to Taylor vortices.

6. Classical Types of Taylor VorticesClassical Types of Taylor Vortices Chaotic Flow Happens Before Fully Turbulence The flow undergoes a series of transitions before it becomes fully turbulent. By using spectral method, there is only one peak at the initial period of the wavy vortex flow, while more peaks will be formed when Re is increasing, this period is called Chaotic Flow until no peaks appear.

7. Terminology k- Thermal Conductivity or circumferential wave number ν - Kinematic Viscosity u θ - Azimuthal Velocity u 0 - Tangential Velocity of Inner Cylinder r- Distance from Centre Axis N-number of cells m-number of azimuthal waves ω -fundamental angular frequency of the wave Radius Ratio η =R 1 /R 2 Gap Width d=b-a Aspect Ratio Г =H/d Re=R 1 Ω d/ ν Axial Wavelength λ =2H/N=(2 Г /N)d Wave Speed s= ω /(m Ω ) Mean Equivalent Conductivity Keq= -h*r*ln(R 1 /R 2 )/k h- Convective Heat Transfer Coefficient R 1 - Radius of Inner Cylinder R 2 - Radius of Outer Cylinder H- height Ω - angular velocity of inner cylinder

8. Important ParametersImportant Parameters  η Critical Reynolds number for forming taylor vortices is strongly influenced by the radius ratio of two cylinders.

9. Important ParametersImportant Parameters  Aspect ratio It is noted that aspect ratio is expected to have little effect on the quantitative behavior of the flow for aspect ratios above 40. [J.A. Cole 1976]

10. Important ParametersImportant Parameters  Axial wavelength & Wave number [D. Coles (1965)] entailed the observation of different flow states with the same Taylor number and concluded that the flow depends not only on Re, but also on previous flow history. The states are defined by number of axial and azimuth waves.

11. Important ParametersImportant Parameters  Wave speed [KIKG,G. P. (1984)] concluded that there was much weaker dependence of wave speed on axial wavelength, azimuthally wave number, and the aspect ratio.

12. Important ParametersImportant Parameters  Flow structure [M. Fenot (2011)] concluded that the structure of the combined flow in annular space hinges not only on the operation point (axial Reynolds and Taylor numbers), but also – and strongly- on geometry and, to a lesser degree, on parietal thermal condition.

13. Mesh Independence Study Objectives & Methodology  After comparing with experiment data, we can find the minimum mesh density that can appropriately simulate the physical phenomenon. With such confidence, a more reasonable mesh structure would be selected as a final reference for the later mesh building.  All the boundary conditions are kept the same with changing the mesh grid numbers only.  The length was cut into half to save computing time. Radial Grids Axial Grids Circle Grids

14. Mesh Independence Study Boundary Conditions  R 1 =0.1906m R 2 =0.2622m H=1.6714m (full length)  Re=4Re c Ω =0.0244 rad/s  End walls are rotating with inner cylinder while outer was fixed  Laminar mode  Fluid: Water (µ=0.001003kg/m-s ρ =998kg/m 3 ν =1.005*e-6 m 2 /s)

15. Mesh Independence StudyMesh Independence Study Circle Grids Radial Grids Axial Grids Cells Numbe r Residual for Continuity Accordingly Wavelength 20060250289.98E-041.569 10060250281.26E-031.575 *Axial grid number here is for full length. In computing convenience, the number was cut into half of that in the form. *The experimental value of wavelength is 1.59 No influence on the cells number for decreasing the circle grid number. So keep it as 100.

16. Mesh Independence StudyMesh Independence Study Circle Grids Radial Grids Axial Grids Cells Numb er Residual for Continuity Accordingly Wavelength 10060120281.38E-031.588 10060180282.42E-031.610 10060250281.26E-031.575 10060400282.05E-031.541 *Axial grid number here is for full length. In computing convenience, the number was cut into half of that in the form. *The experimental value of wavelength is 1.59 No influence on the cells number for an interval of circle grid numbers.

17. Mesh Independence StudyMesh Independence Study 100 60 120100 60 180 100 60 250 100 60 400 Cross Section View with MeshCross Section View with Mesh

18. Mesh Independence StudyMesh Independence Study  According to the above figures, simulation shall be kept reasonable with each cell shares at least 3 grid axial. In order to achieve relatively accurate result, the one with axial grid number of 400 is selected as the reference.

19. Mesh Independence StudyMesh Independence Study Circle Grids Radial Grids Axial Grids Cells Numb er Residual for Continuity Accordingly Wavelength 10080180285.21E-02N/A 10060180282.42E-031.610 10050180243.03E-031.852 10040180245.21E-02N/A *Axial grid number here is for full length. In computing convenience, the number was cut into half of that in the form. *The experimental value of wavelength is 1.59. The minimum for the radial grid number is 60.

20. Mesh Independence StudyMesh Independence Study  According to the real computing condition, the mesh with circle 100, radial 60 and axial 400 (full length) is good enough which can be regarded as the reference for later mesh building. Conclusion

21. Taylor-Couette Validation Objectives & Methodology  After comparing with the experiment data from J.E. Burkhalter & E.L. koschmieder (1974) and check wavelength versus taylor number. We shall confirm that fluent is able to simulate the Taylor- Couette phenomena. λ

22. Taylor-Couette Validation Boundary Conditions  Since author have mentioned that neither the end wall nor the column length have significant influence on the experiment result, so we kept the same boundary condition with former mesh independence study, only changing the Taylor number which is exact a non- dimension value.

23. Taylor-Couette ValidationTaylor-Couette Validation Wavelength T/Tc Full Length Typical wavelengths after sudden starts

24. Taylor-Couette Validation Conclusion  From above figure, we tried both full length and half length geometries to run cases with different taylor numbers. The results under 8Tc are well fit with the experiment data. Those above 8Tc are absolutely among the wavy taylor vortices period which will possibly cause the unstable measurement for the wavelength. However, the discrepancy is acceptable

25. Wavy Taylor Validation Objectives & Methodology  Taking KIKG,G. P., Lr, Y., LEE, W., SWINNEY, H. L. & MARCUS, P.S. 1984 Wave speeds in wavy Taylor-vortex flow. J. Fluid Mech. 141, 365-390. as the reference paper to compare with. Check wave speed versus different radius ratio which is the most significant factor.  Set measure point located in the gap centre from 2D cross section view to record the z-velocity during the time interval which can derive the fundamental frequency of the azimuthal waves. m can be observed by applying Tecplot to stretch out the cylinder.

26. Wavy Taylor Validation Boundary Conditions *All the meshes are built with reference of “100 60 400” which has been mentioned in the former part. η (R1/R2)R1(cm)R2(cm)H(cm)Re(11Rc) Ω (rad/s) Upper Boundcircumradialaxial 0.8682.2052.54010.0501266.117.222Free10030250 0.9002.2862.5407.6201447.625.050Free10025190 0.9505.6495.9468.9102036.112.194Free20025200  Laminar mode  Fluid: Water (µ=0.001003kg/m-s ρ =998kg/m3 ν =1.005*e-6 m2/s)  Keep Г =30, Re=11Rc, λ /d=2.4

27. Wavy Taylor ValidationWavy Taylor Validation η (R1/R2)R1(cm)R2(cm)H(cm)Re(11Rc) Ω (rad/s) Upper Boundcircumradialaxial 0.8682.2052.54010.0501266.117.222Free10030250 Fundamental angular frequency ω =17.279 s= ω /(m Ω )=0.334

28. Wavy Taylor ValidationWavy Taylor Validation Two fundamental frequencies ω =27.227 s= ω /(m Ω )=0.362 η (R1/R2)R1(cm)R2(cm)H(cm)Re(11Rc) Ω (rad/s) Upper Boundcircumradialaxial 0.9002.2862.5407.6201447.625.050Free10025190

29. Wavy Taylor ValidationWavy Taylor Validation Fundamental angular frequency ω =50.265 s= ω /(m Ω )=0.458 η (R1/R2)R1(cm)R2(cm)H(cm)Re(11Rc) Ω (rad/s) Upper Boundcircumradialaxial 0.9505.6495.9468.9102036.112.194Free20025200

30. Wavy Taylor Validation Conclusion η (a/b)Computed S1Measured S1 0.8680.3340.320±0.005 0.9000.3620.360±0.010 0.9500.4580.450±0.001 The difference is located in the reasonable region of uncertainty Need to be calculated longer. Dependence of s1 on radius ratio

31. Turbulent Validation Objectives & Methodology Comparison of normalized mean angular momentum pro fi les between present simulation (Re=8000) and the experiment of Smith & Townsend (1982).  Using two models (k-epsilon and k- omega) to compare with the experiment data (left), after which an appropriate model would be selected for later calculation.  Couple of measure points are located in the midline across the gap from 2D view of the cross section. They are applied to record the tangential velocity during time interval and calculate the average value.

32. Turbulent Validation Boundary Conditions R 1 = 0.1525 m R 2 = 0.2285 m Ω = 22.295 rad/s (Re=17295) H = 1.80 m End walls are free surfaces k- epsilon and k- omega were chosen to compare Fluid: Water (µ=0.001003kg/m-s ρ =998kg/m3 ν =1.005*e-6 m2/s) Mesh Density Axial = 400 Circle = 100 Radial = 60

33. Turbulent Validation Comparing with Experiment Data

34. Turbulent Validation Conclusion  Flow time interval is not enough ΔT epsilon =27.68s ΔT omega =20.48s  Sampling frequency f experiment =10kHz f simulation =200Hz  Mesh density Tip: W. M. J. Batten used k-epsilon as the turbulent model

35. Thermal Validation Objectives & Methodology Variation of mean equivalent conductivity with Reynolds number for different Grashof numbers  By comparing with experiment data from K.S. Ball (1989), we would confirm the capability of fluent on simulating the heat transfer for taylor- couette flow.  We simplified the condition by ignore the effect of both conduction and radiation which will cause at most 5% error but saving much calculating time. Average heat transfer coefficient in the inner cylinder surface can be directly achieved from fluent.

36. Thermal Validation Boundary Conditions K eq = -h*r*ln(R 1 /R 2 )/k Re = Ω* (R 1 -R 2 )*R 1 /ν R 1 = 1.252 cm R 2 = 2.216 cm H = 50.64 cm Gr= 1000 ΔT= 7.582 K Ti = 293K To= 300.582K End walls are fixed and insulated Fluid: Air (k=0.0257w/m-k β=3.43*e -3 1/k ν=15.11*e -6 m 2 /s) Re=[40 120 280] Ω=[5.008 15.023 35.054] rad/s Since for η=0.565 Re c = 70, All the three cases are in laminar mode. Mesh Density Axial = 1000 Circle = 100 Radial = 60

37. Comparing with Experiment Data Re 2 h(w/m 2 k)k eq R. KediaExperiment DataResidue 16003.6001.001N/A1.0809.8e-04 64004.6361.289N/A1.4501.4e-03 144006.2061.7251.651.5002.0e-03 400008.5452.3762.301.7501.7e-03 7840010.2282.8452.702.1201.8e-03

38. Comparing with Experiment Data

39. Possible Reasons for Difference  Boundary condition set-up ideal gas, pressure based, real apparatus error (axial temperature gradient, end walls effect) “This discrepancy arises because of the variation of axial wave length with axial distance, which results from the thermal conditions at the ends of the experimental apparatus.” [R. Kedia (1997)]

40. Simple Model TestSimple Model Test R 1 = 96.85 mm R 2 = 97.5 mm Height = 140 mm Q=4 L/min V in = 0.000168 m/s T in = 308K T out = 551K Ω=29.311 rad/s End walls are fixed and insulated Measure points are located in the vertical lines close to the inner cylinder. Since for η=0.975 Re c = 260.978, In this case Re=1837.075 So, it is in laminar mode.

41. Plans for Next PeriodPlans for Next Period  Keep running both of the turbulent cases  Finish the thermal validation  Repeat Taylor-couette validation with full length  Wavy validation should be finished with running 0.95 case long enough  More validation of the thermal part (optional)  Keep turbulent case running  Finish simple model test  Check geometry related paper