content classical types of taylor vortices important parameters mesh independence study ...

Summary for Validation Period

Qian Wentao

24/05/2011

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


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.


Taylor vortex forms when Re exceeds Rec

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.


Wavy taylor vortex forms when Re exceeds Rec2

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.


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.

Terminology

k- Thermal Conductivity or circumferential wave numberν - Kinematic Viscosityuθ- Azimuthal Velocityu0- Tangential Velocity of Inner Cylinderr- Distance from Centre Axis N-number of cellsm-number of azimuthal wavesω-fundamental angular frequency of the wave

Radius Ratio η=R1/R2 Gap Width d=b-aAspect Ratio Г=H/d Re=R1Ωd/νAxial Wavelength λ=2H/N=(2Г/N)dWave Speed s=ω/(m Ω)Mean Equivalent Conductivity Keq= -h*r*ln(R1/R2)/k

h- Convective Heat Transfer CoefficientR1- Radius of Inner CylinderR2- Radius of Outer CylinderH- heightΩ- angular velocity of inner cylinder

Important Parameters η

Critical Reynolds number for forming taylor vortices is strongly influenced by the radius ratio of two cylinders.


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]


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.


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.

Important 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.


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


Boundary Conditions

R1=0.1906m R2=0.2622m H=1.6714m (full length)

Re=4Rec 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/m3 ν=1.005*e-6 m2/s)


Circle Grids

Radial Grids

Axial Grids

Cells Numb

erResidual for Continuity

Accordingly Wavelength

200 60 250 28 9.98E-04 1.569100 60 250 28 1.26E-03 1.575

*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.


Circle Grids

Radial Grids

Axial Grids

Cells Numb



100 60 120 28 1.38E-03 1.588100 60 180 28 2.42E-03 1.610100 60 250 28 1.26E-03 1.575100 60 400 28 2.05E-03 1.541

*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.


100 60 120 100 60 180 100 60 250 100 60 400

Cross Section View with Mesh


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.


Circle Grids

Radial Grids

Axial Grids

Cells Numb



100 80 180 28 5.21E-02 N/A100 60 180 28 2.42E-03 1.610100 50 180 24 3.03E-03 1.852100 40 180 24 5.21E-02 N/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.


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


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


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.

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.21

2

4

8

16

32

64

J.E. Burkhalter & E.L. koschmieder (1974)Simulation Data


Wavelength

T/T

c

Full Length

Typical wavelengths after sudden starts

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

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.

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 Boun

dcircu

m radial axial

0.868 2.205 2.54010.05

0 1266.1 17.222 Free 100 30 2500.900 2.286 2.540 7.620 1447.6 25.050 Free 100 25 1900.950 5.649 5.946 8.910 2036.1 12.194 Free 200 25 200

Laminar mode


Keep Г=30, Re=11Rc, λ/d=2.4


η(R1/R2)

R1(cm)

R2(cm) H(cm)

Re(11Rc)

Ω(rad/s)

Upper Bound circum

radial

axial

0.868 2.205 2.54010.05

0 1266.1 17.222 Free 100 30 250

Fundamental angular frequencyω=17.279s=ω/(m Ω)=0.334


Two fundamental frequenciesω=27.227s=ω/(m Ω)=0.362

η(R1/R2)

R1(cm)

R2(cm) H(cm)

Re(11Rc)

Ω(rad/s)

Upper Bound circum

radial

axial

0.900 2.286 2.540 7.620 1447.6 25.050 Free 100 25 190


Fundamental angular frequencyω=50.265s=ω/(m Ω)=0.458

η(R1/R2)

R1(cm)

R2(cm) H(cm)

Re(11Rc)

Ω(rad/s)

Upper Bound circum

radial

axial

0.950 5.649 5.946 8.910 2036.1 12.194 Free 200 25 200

Wavy Taylor ValidationConclusion

η(a/b) Computed S1

Measured S1

0.868 0.334 0.320±0.005

0.900 0.362 0.360±0.010

0.950 0.458 0.450±0.001

The difference is located in the reasonable region of uncertainty

Need to be calculated longer.

Dependence of s1 on radius ratio

Turbulent Validation Objectives & Methodology

Comparison of normalized mean angular momentum profiles between presentsimulation (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.

Turbulent Validation Boundary Conditions

R1 = 0.1525 mR2 = 0.2285 mΩ = 22.295 rad/s (Re=17295)H = 1.80 m

End walls are free surfaces

k- epsilon and k- omega were chosen to compare


Mesh DensityAxial = 400Circle = 100Radial = 60

Turbulent Validation Comparing with Experiment Data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.3

0.4

0.5

0.6

0.7

0.8

Experiment DataSimulation Data k-epsilonSimulation Data k-omega

z/d

U*r

/(U

1*R

1)

Turbulent Validation Conclusion

Flow time interval is not enough

ΔTepsilon=27.68s ΔTomega=20.48s

Sampling frequency

fexperiment=10kHz fsimulation=200Hz

Mesh density

Tip: W. M. J. Batten used k-epsilon as the turbulent model

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.

Thermal Validation Boundary Conditions

Keq= -h*r*ln(R1/R2)/kRe = Ω* (R1-R2)*R1/ν

R1 = 1.252 cmR2 = 2.216 cmH = 50.64 cmGr= 1000 ΔT= 7.582 KTi = 293K To= 300.582KEnd walls are fixed and insulated

Fluid: Air (k=0.0257w/m-k β=3.43*e-3 1/k ν=15.11*e-6 m2/s)

Re=[40 120 280]Ω=[5.008 15.023 35.054] rad/s

Since for η=0.565 Rec= 70,All the three cases are in laminar mode.

Mesh DensityAxial = 1000Circle = 100Radial = 60

Comparing with Experiment Data

Re2 h(w/m2k) keq R. Kedia Experiment Data

Residue

1600 3.600 1.001 N/A 1.080 9.8e-04

6400 4.636 1.289 N/A 1.450 1.4e-03

14400

6.206 1.725 1.65 1.500 2.0e-03

40000

8.545 2.376 2.30 1.750 1.7e-03

78400

10.228 2.845 2.70 2.120 1.8e-03

Comparing with Experiment Data

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)]

Simple Model Test

R1 = 96.85 mmR2 = 97.5 mmHeight = 140 mm Q=4 L/min Vin= 0.000168 m/sTin = 308K Tout= 551KΩ=29.311 rad/sEnd walls are fixed and insulated

Measure points are located in the vertical lines close to the inner cylinder.

Since for η=0.975 Rec= 260.978,In this case Re=1837.075So, it is in laminar mode.

Plans 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

content classical types of taylor vortices important parameters mesh independence study ...

Documents

taylor number

turbulencethe flow

combined flow

couette laminar flow

aswavy vortex flow

angular speed

number of axial

wave radius ratio