Size scaling relation of velocity field in granular flows and the Beverloo law

Gaoke Hu; Ping Lin; Yongwen Zhang; Liangsheng Li; Lei Yang; Xiaosong Chen

doi:10.1007/s10035-019-0872-z

Size scaling relation of velocity field in granular flows and the Beverloo law

Gaoke Hu, Ping Lin, Yongwen Zhang, Liangsheng Li, Lei Yang, Xiaosong Chen

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow v _z depends on the distance from the hopper’s center r and the height above the aperture z and vz=vz(r,z;R). We propose that the scaled vertical velocity vz(r,z;R)/vz(0,0;R) is a function of scaled variables r/ R _r and z/ R _z , where R _r = R- 0.5 d and R _z = R- k ₂ d with the granule diameter d and a parameter k ₂ to be determined. After scaled by vz2(0,0;R)/Rz, the effective acceleration aeff(r,z;R) derived from v _z is a function of r/ R _r and z/ R _z also. The boundary condition aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to vz(0,0;R)∝g(R-k2d)1/2. Our simulations using the discrete element method and GPU program in the three-dimensional and the two-dimensional hoppers confirm the size scaling relations of vz(r,z;R) and vz(0,0;R). From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper W∝g(R-0.5d)D-1(R-k2d)1/2, which agrees with the Beverloo law at R≫ d. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.

Original language	English
Article number	21
Journal	Granular Matter
Volume	21
Issue number	2
DOIs	https://doi.org/10.1007/s10035-019-0872-z
State	Published - 1 May 2019
Externally published	Yes

Bibliographical note

Funding Information:
Acknowledgements This work is supported by the National Magnetic Confinement Fusion Science Program of China under Grant No. 2014GB104002, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA03030100, and the National natural Science Foundation of China under Grant No. 11421063.

Funding Information:
This work is supported by the National Magnetic Confinement Fusion Science Program of China under Grant No. 2014GB104002, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDA03030100, and the National natural Science Foundation of China under Grant No. 11421063.

Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

Beverloo law
Granular flow
Size scaling relation

Access to Document

10.1007/s10035-019-0872-z

Cite this

@article{fee579a00c92499f8af1905e2a4f108e,

title = "Size scaling relation of velocity field in granular flows and the Beverloo law",

abstract = " In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow v z depends on the distance from the hopper{\textquoteright}s center r and the height above the aperture z and vz=vz(r,z;R). We propose that the scaled vertical velocity vz(r,z;R)/vz(0,0;R) is a function of scaled variables r/ R r and z/ R z , where R r = R- 0.5 d and R z = R- k 2 d with the granule diameter d and a parameter k 2 to be determined. After scaled by vz2(0,0;R)/Rz, the effective acceleration aeff(r,z;R) derived from v z is a function of r/ R r and z/ R z also. The boundary condition aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to vz(0,0;R)∝g(R-k2d)1/2. Our simulations using the discrete element method and GPU program in the three-dimensional and the two-dimensional hoppers confirm the size scaling relations of vz(r,z;R) and vz(0,0;R). From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper W∝g(R-0.5d)D-1(R-k2d)1/2, which agrees with the Beverloo law at R≫ d. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.",

keywords = "Beverloo law, Granular flow, Size scaling relation",

author = "Gaoke Hu and Ping Lin and Yongwen Zhang and Liangsheng Li and Lei Yang and Xiaosong Chen",

note = "Publisher Copyright: {\textcopyright} 2019, Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2019",

month = may,

day = "1",

doi = "10.1007/s10035-019-0872-z",

language = "אנגלית",

volume = "21",

journal = "Granular Matter",

issn = "1434-5021",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - Size scaling relation of velocity field in granular flows and the Beverloo law

AU - Hu, Gaoke

AU - Lin, Ping

AU - Zhang, Yongwen

AU - Li, Liangsheng

AU - Yang, Lei

AU - Chen, Xiaosong

PY - 2019/5/1

Y1 - 2019/5/1

N2 - In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow v z depends on the distance from the hopper’s center r and the height above the aperture z and vz=vz(r,z;R). We propose that the scaled vertical velocity vz(r,z;R)/vz(0,0;R) is a function of scaled variables r/ R r and z/ R z , where R r = R- 0.5 d and R z = R- k 2 d with the granule diameter d and a parameter k 2 to be determined. After scaled by vz2(0,0;R)/Rz, the effective acceleration aeff(r,z;R) derived from v z is a function of r/ R r and z/ R z also. The boundary condition aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to vz(0,0;R)∝g(R-k2d)1/2. Our simulations using the discrete element method and GPU program in the three-dimensional and the two-dimensional hoppers confirm the size scaling relations of vz(r,z;R) and vz(0,0;R). From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper W∝g(R-0.5d)D-1(R-k2d)1/2, which agrees with the Beverloo law at R≫ d. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.

AB - In a hopper with cylindrical symmetry and an aperture of radius R, the vertical velocity of granular flow v z depends on the distance from the hopper’s center r and the height above the aperture z and vz=vz(r,z;R). We propose that the scaled vertical velocity vz(r,z;R)/vz(0,0;R) is a function of scaled variables r/ R r and z/ R z , where R r = R- 0.5 d and R z = R- k 2 d with the granule diameter d and a parameter k 2 to be determined. After scaled by vz2(0,0;R)/Rz, the effective acceleration aeff(r,z;R) derived from v z is a function of r/ R r and z/ R z also. The boundary condition aeff(0,0;R)=-g of granular flows under earth gravity g gives rise to vz(0,0;R)∝g(R-k2d)1/2. Our simulations using the discrete element method and GPU program in the three-dimensional and the two-dimensional hoppers confirm the size scaling relations of vz(r,z;R) and vz(0,0;R). From the size scaling relations, we obtain the mass flow rate of D-dimensional hopper W∝g(R-0.5d)D-1(R-k2d)1/2, which agrees with the Beverloo law at R≫ d. It is the size scaling of vertical velocity field that results in the dimensional R-dependence of W in the Beverloo law.

KW - Beverloo law

KW - Granular flow

KW - Size scaling relation

UR - http://www.scopus.com/inward/record.url?scp=85062207673&partnerID=8YFLogxK

U2 - 10.1007/s10035-019-0872-z

DO - 10.1007/s10035-019-0872-z

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85062207673

SN - 1434-5021

VL - 21

JO - Granular Matter

JF - Granular Matter

IS - 2

M1 - 21

ER -

Size scaling relation of velocity field in granular flows and the Beverloo law

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this