DEVELOPMENT AND VALIDATION OF A NEW HISTORY FORCE MODEL WITH COLLISION TREATMENT

2015 NETL WORKSHOP ON MULTIPHASE FLOW SCIENCE

Husam Elghannay &Danesh Tafti

Department of Mechanical EngineeringVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

12015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Outline

• Motivation• Model formulation– Rep <<1

– Finite Rep

– Collision handling

• Validation– Settling spheres– Bouncing Spheres

• Computational resources• Summary and Conclusions

22015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Particle EOM

�⃗� 𝐹𝑆=1𝜌𝑝

(− �⃗�𝑝+�⃗�𝜏 )= 𝜌 𝑓

𝜌𝑝(𝑑�⃗�𝑑𝑡−�⃗�); drag coefficient

�⃗� 𝑔𝑟𝑎𝑣𝑖𝑡𝑦=�⃗�

�⃗� 𝑎𝑑𝑑𝑚=𝑐∀

𝜌 𝑓

𝜌𝑝( 𝑑 (�⃗�−�⃗�𝑝)

𝑑𝑡 )

�⃗� 𝑙𝑖𝑓𝑡 , h𝑠 =5.2√𝜐 𝑓 𝜌 𝑓 𝑠𝑖𝑗

𝜌𝑝𝑑𝑝 (𝑠𝑙𝑘 𝑠𝑘𝑙 )14

(�⃗�−�⃗�𝑝 )×𝐶1

Drag; viscous dissipation exerted by fluid

Gravity; particle weight

Added mass; acceleration of surrounding fluid by the particle

History force; accounts for temporal evolution of viscous region in the vicinity of the particle

Fluid Stress; Effects of gradients of fluid occupied by particle

Lift force; due to velocity gradients across particles

�⃗� 𝐻𝑖𝑠𝑡𝑜𝑟𝑦=−18𝜇𝑓

𝑑❑2

𝜌 𝑓

𝜌𝑝∫0

𝑡

𝐾 (𝑡−𝜏 )𝑑 𝑉 𝑟𝑒𝑙

𝑑𝜏𝑑𝜏

�⃗� 𝑙𝑖𝑓𝑡 , 𝑠𝑝𝑖𝑛=−0.75𝜌 𝑓

𝜌𝑝( �⃗�× (�⃗�−�⃗�𝑝 ))×𝐶2

32015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

History Force Model

• Formula• Expensive to calculate

– E.g. expand for three time steps (Rostami et al. 2006)

• Memory storage;– 1.2GB are required to store

4-byte-digit of relative velocity information for 100k particles for 1000 time steps (Dorgan & Loth 2007)

• Can make the calculation impractical with large number of particles

=

42015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Reduction Techniques

• Gonzalez et al. (2006)– Use of fractional-

derivative approach– Concept of memory time

period– Saves 10-30% of

simulation time

• Dorgan & Loth (2007)– Window Model– Can saves up to 2-

Orders of magnitude of CPU-time per time step

𝐾 𝑤𝑖𝑛𝑑𝑜𝑤={𝐾 𝐵𝑎𝑠𝑠𝑒𝑡

0𝑓𝑜𝑟

𝑡−𝜏h<𝜏<𝑡𝜏<𝑡−𝜏h

5

Log(th) for Rep=1000

-2

-2

-1/2

2015 NETL Workshop on Multiphase Flow Science, August, 2015 Morgantown, WV, 26508

Motivation and Objectives

• Goal: To develop an efficient history force model to be used in CFD-DEM

– Should not be expensive since large number of particles is usually involved in simulating industrial and natural systems

– Used for finite Reynolds numbers

– Proper handling of particle-wall particle-particle interactions

62015 NETL Workshop on Multiphase Flow Science,August 12, 2015 Morgantown, WV, 26508

Model Formulation (Rep <<1)

• L

• For a single current time step

• The cumulative non-decaying sum of instantaneous time steps (=const.)

• Assume the relation to be the same if the acceleration is not constant 72015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Decay Function• g(n)= , decreases with

increase in n, will be termed “Decay function”

• Use fit functions of the function

• For comparison f(n)=n-1/2 is shown on same plot

82015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Model Formulation (Finite Rep)

• Not as straight forward as for Rep <<1

• Suggested fit function when history time is exceeded;– g2(n)= 1.53.n-2

• Final form𝐹 h=𝑔1 (𝑛 ) . (∑

𝑖=1

𝑛h

𝐹𝐵 ,∆𝑡+ ∑𝑖=1

𝑛𝑡𝑜𝑡 −𝑛h

𝐹𝑀𝐴 , ∆𝑡) .( 𝑛h

𝑛𝑡𝑜𝑡)+𝑔2 (𝑛) .(∑

𝑖=1

𝑛h

𝐹𝐵 , ∆𝑡+ ∑𝑖=1

𝑛 𝑡𝑜𝑡−𝑛h

𝐹𝑀𝐴 , ∆𝑡) .(𝑛𝑡𝑜𝑡−𝑛h

𝑛𝑡𝑜𝑡)

92015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Collision Handling

• Collision force is the dominant force when particles are interacting

• Strategy :– Apply only the instantaneous History

force during collision– Reset the cumulative history force to

zero (correlation with history is lost) – Use Mei correction for non-zero initial

condition once the collision is completed

102015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

(S

~2

.47

, R

ep ~

1-6

) (

S~

3.7

, R

ep ~

30

-18

0)

11

Validation: Settling spheres

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

( R

ep ~

30

, S

~2

.5-9

.2)

12

Validation: Settling spheres

(S

~1

.16

, R

ep ~

2-1

2)

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

(Rep,i~15-400, S~8)

13

Validation: Bouncing spheres

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

14

Validation: Bouncing spheres(Rep,i~100, S~1.5-16)

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

, S~8

Computational Resources

• Two Mock sub-routines to calculate the History force using Window Model and traditional calculation using Basset Kernel (less expensive)

• Moorman RUN#27 (Rep,term ~29) was tested

152015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Summary• A new formulation of History force along w/

method of handling collisions was presented and validated for large range of Rep and density ratios

• The model compares well with experimental measure -ments of settling spheres and shows reasonably good agreement with bouncing sphere experiments

• The use of the model and the fit function reduces both memory resources and CPU time required to calculate the history force

162015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Thank You

172015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Comments?

Validation: Settling spheres

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

Run # Rep,term d(mm) rp (kg/m3) rf (kg/m3) n f (m2/s) td(s)

Moorman (1955)10 31.5 11.1 7782.2 850.4 2.7e-4 0.4553

19 181 15.88 3076.8 834.9 7.2e-5 3.502

21 67 9.51 3076.8 834.9 7.2e-5 1.256

22 28 6.37 3076.8 834.9 7.2e-5 0.56363

27 29 12.71 3076.8 1247.2 1.48645e-4

1.0868

29 6 6.37 3076.8 1247.2 1.48645e-4

0.273

30 3.8 6.37 3076.8 1252.4 1.877e-4 0.216

31 0.9 6.37 3076.8 1257.5 4.4872e-4 0.0904

Mordant and Pinton (2000)CASE-1 41 0.5 2560 1000 1e-6 0.25

CASE-4 280 0.8 7710 1000 1e-6 0.64

Ten Cate et. al. (2002)E1 1.5 15 1120 970 3.85e-4 0.585

E2 4.1 15 1120 965 2.2e-4 1.024

E3 11.6 15 1120 962 1.17e-4 1.91518

Validation (Mordant and Pinton)

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

19

Validation: Bouncing spheres

2015 NETL Workshop on Multiphase Flow Science, August 12, 2015 Morgantown, WV, 26508

• Gondret et al. (2001) experiments were selected• Drag Model used is the same as of Gondret• Soft sphere collision model was used (k=800 N/m)• Spheres released from height equal to rebound height

and their restitution coefficient was set to 1• Forces where activated just before the collision is

completedFig # Rep,i dp(mm) rs (kg/m3) rf (kg/m3) m f

(m2/s)

td (s) Rep,reb

Gondret10A 394 5 7800 920 0.005 4.6 36910B-11B

106 6 7800 953 0.021.7154

83.48

10C 55 4 7800 953 0.02 0.7624 3910D 15 6 7800 965 0.01 0.3474 711A 108 5 14970 953 0.02 1.19125 96.211C 119 6 2500 935 0.01 3.366 45.511D 91 5 1410 920 0.005 4.6 14.8

20