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Time Scales and Turbulent Spectra above the Base of Stirred Vessels from Large Eddy Simulations

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Abstract

Single-phase Large Eddy Simulations (LESs) have been conducted with M-Star CFD software to compute spectra and time scales of the turbulent flow field at positions above the base of a stirred tank as these time scales may be important to the application of solids suspension. Since LESs do not resolve the entire turbulence spectrum with an unknown effect on computing time scales, simulation aspects such as spatial resolution, tank size and the value of the Smagorinsky constant are investigated. The simulations were conducted for the case of a 108 mm diameter axial flow wide blade hydrofoil, Philadelphia Mixing Solutions Ltd. 3MHS39, in a 295 mm diameter vessel at specific power inputs in the range from 0.03 to 0.43 W/kg. These specific power inputs pertain to the energy required to suspend 600 to 10,000 μm glass spheres at 0.9% (v/v). The focus of the study was on the base of the vessel near the vessel corners where solid particles are observed to reside before suspension in solid-liquid mixing application. Integral and Taylor micro time scales as well as 1-D energy spectra were extracted, for all three velocity components, from the temporal autocorrelation function of the resolved velocity traces at various positions. Radial profiles are presented for these 1-D time scales, for the 3-D integral time scale, and for the resolved time scale k/ε.

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  • 11 February 2020

    The correct figures are presented in this correction article.

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Acknowledgments

The first author wishes to acknowledge John Thomas, the CEO and creator of M-StarCFD, for his assistance and tutelage in the methods of LES and Lattice-Boltzmann and their application to this research. The first author wishes to acknowledge Philadelphia Mixing Solutions ltd. for supporting the work conducted herein.

Author information

Authors and Affiliations

  1. Philadelphia Mixing Solutions Ltd., Palmyra, PA, USA

    Jason J. Giacomelli

  2. Bernal Institute, School of Engineering, University of Limerick, Limerick, Ireland

    Jason J. Giacomelli & Harry E. A. Van den Akker

Authors
  1. Jason J. Giacomelli
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  2. Harry E. A. Van den Akker
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Corresponding author

Correspondence to Jason J. Giacomelli.

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Conflict of Interests

The first author wishes to disclose that the work presented in this research was funded by Philadelphia Mixing Solutions, ltd. The first author is both a full-time employee of Philadelphia Mixing Solutions, ltd. and a part-time student at the University of Limerick, Ireland. Both authors declare no conflict of interest.

Nomenclature

C S

Smagorinsky Constant

–

dt

Simulation Time Step

s

d p43

Mean Particle Diameter

μm

D

Impeller Diameter

m

E(f)

Kinetic Energy Spectrum

m2/s2

f

Frequency

Hz

H

Liquid Height

m

k res

Resolved Kinetic Energy

m2/s2

ℓ S

Smagorinsky Mixing Length

m

N

Impeller Speed

Revs/s or Revs/min

N P

Impeller Power Number, \( \frac{P}{\rho {N}^3{D}^5} \)

–

P

Impeller Power Input

W

r

Radial Direction (cylindrical)

–

\( \overline{\mathbf{\mathcal{S}}} \)

Total Resolved Strain Rate, \( \sqrt{2{\overline{S}}_{ij}{\overline{S}}_{ij}} \)

s−1

\( {\overline{S}}_{ij} \)

Resolved Strain Rate Tensor, \( \frac{1}{2}\left(\frac{{\partial u}_i}{{\partial x}_j}+\frac{{\partial u}_j}{{\partial x}_i}\right) \)

s−1

t

Time

s

T

Tank Diameter

m

u

(Instantaneous) Velocity

m/s

u ′

Fluctuating Velocity

m/s

U

Mean Velocity

m/s

u res

Resolved Velocity

m/s

u SGS

Subgrid Scale Velocity

m/s

u tip

Impeller Tip Velocity, πND

m/s

Re

Reynolds Number, \( \frac{N{D}^2}{\nu } \)

–

x

East/West Direction (rectangular)

–

y

Axial Direction (cylindrical/rectangular)

–

z

North/South Direction (rectangular)

–

Δ

Lattice Size

m or mm

\( \overline{\varepsilon} \)

Average Specific Power Input

m2/s3

ε Local

Local Specific Power Input

m2/s3

ε res

Resolved Energy Dissipation Rate

m2/s3

η k

Kolmogorov Scale, \( {\left(\frac{\nu^3}{\varepsilon}\right)}^{1/4} \)

m

θ

Tangential Direction (cylindrical)

–

Λ

Impeller Torque

J

ν

Kinematic Molecular Viscosity

m2/s

ν SGS

Kinematic Subgrid Scale Viscosity

m2/s

\( \overline{\nu} \)

Effective Kinematic Viscosity

m2/s

ρ(τ)

Autocorrelation Function

–

ρ

Fluid Density

kg/m3

τ

Lag Time

s

τ 0, i

Integral Time Scale

s

τ λ

Taylor Micro Time Scale

s

τ f, res

Resolved Fluid Time Scale

s

τ N

Impeller Rotational Time Scale

s

Additional information

The original version of this article was revised and the correct Figs. 1, 3 and 4 inserted.

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Giacomelli, J.J., Van den Akker, H.E.A. Time Scales and Turbulent Spectra above the Base of Stirred Vessels from Large Eddy Simulations. Flow Turbulence Combust 105, 31–62 (2020). https://doi.org/10.1007/s10494-019-00095-z

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