## Abstract

An analytical expression for turbulent kinematic viscosity ($νt$), based solely on the hydraulic Reynolds number (Re), was derived and evaluated. The analytical expression is valid for the fast estimation of $νt$ for internal, isotropic, fully developed flows. The expression was compared with experimental and simulation data for air, water, and liquid sodium, and was shown to provide reasonable values for 2100 ≤ Re ≤ 3.6 $×$ 106 and Prandtl number (Pr) range of 0.0107 ≤ Pr ≤ 5.65. In addition, new expressions suitable for the central portion of internal flows, away from the wall, were derived for the turbulent Reynolds number ($ReT$), showing its relationship to Re, as well as to the ratio of $νt$ and the molecular kinematic viscosity ($ν$).

## References

1.
Canuto
,
V. M.
,
Goldman
,
I.
, and
Chasnov
,
J.
,
1988
,
Turbulent Viscosity. Astron. Astrophys.
200, pp.
291
300
.
2.
Absi
,
R.
,
2009
, “
A Simple Eddy Viscosity Formulation for Turbulent Boundary Layers Near Smooth Walls
,”
C. R. Mec.
,
337
(
3
), pp.
158
165
.10.1016/j.crme.2009.03.010
3.
Absi
,
R.
,
2019
, “
Eddy Viscosity and Velocity Profiles in Fully-Developed Turbulent Channel Flows
,”
Fluid Dyn.
,
54
(
1
), pp.
137
147
.10.1134/S0015462819010014
4.
Noskov
,
V.
,
Denisov
,
S.
,
Stepanov
,
R.
, and
Frick
,
P.
,
2012
, “
Turbulent Viscosity and Turbulent Magnetic Diffusivity in a Decaying Spin-Down Flow of Liquid Sodium
,”
Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
,
85
(
1
), p.
9
.10.1103/PhysRevE.85.016303
5.
Rodriguez
,
S.
,
2019
,
Applied Computational Fluid Dynamics and Turbulence Modeling
,
Springer
,
Switzerland
, p.
301
6.
Wilcox
,
D. C.
,
2006
,
Turbulence Modeling for CFD
,
DCW Industries Incorporated
CA
, p.
522
.
7.
Kolmogorov
,
A. N.
,
1941
, “
Equations of Turbulent Motion in an Incompressible Fluid
,”
,
30
(
4
), pp.
299
303
.
8.
Taylor, G. I., 1935, “Statistical Theory of Turbulence,”
Proc. R. Soc. London. Ser. A-Math. Phys. Sci
., 151(873), pp. 421–444.10.1098/rspa.1935.0158
9.
Prandtl
,
L.
,
1945
, “Uber Ein Neues Formel-System Fur Die Ausgebildete Turbulenz,”
Nachr. Akad. Wiss, Gottin. Math, Phys. Kl
, 1945, pp. 874–887.10.1007/978-3-662-11836-8_72
10.
Sandborn
,
V. A.
,
1955
, “
Experimental Evaluation of Momentum Terms in Turbulent Pipe Flow
,”
NACA
Tech. Note 3266, Washington, DC, p.
40
, Report No. 3266.https://ntrs.nasa.gov/citations/19930084016
11.
Minin
,
O.
, and
Minin
,
I.
,
2011
,
Computational Fluid Dynamics: Technologies and Applications
,
InTech
,
Rijeka, Croatia
, p.
375
.https://www.intechopen.com/books/161
12.
Russo
,
F.
, and
Basse
,
N. T.
,
2016
, “Scaling of Turbulence Intensity for Low-Speed Flow in Smooth Pipes,”
Flow Meas. Instrum.
, 52, pp. 101–114.10.1016/j.flowmeasinst.2016.09.012
13.
Basse
,
N. T.
,
2017
, “Turbulence Intensity and the Friction Factor for Smooth-and Rough-Wall Pipe Flow,”
Fluids
, 2(2), p. 30.10.3390/fluids2020030
14.
de Karman
,
T.
, and
Howarth
,
L.
,
1938
, “
On the Statistical Theory of Isotropic Turbulence
,”
Proc. R. Soc. London. Ser. A-Math. Phys. Sci.
,
164
(
917
), pp.
192
215
.10.1098/rspa.1938.0013
15.
Kawahara
,
G.
,
2009
, “
Theoretical Interpretation of Coherent Structures in Near-Wall Turbulence
,”
Fluid Dyn. Res.
,
41
(
6
), p.
064001
.10.1088/0169-5983/41/6/064001
16.
Pourghasemi
,
M.
,
Fathi
,
N.
, and
Rodriguez
,
S.
,
2021
, “
Numerical Study on Flow and Heat Transfer of Water and Liquid Metals Within Micro-Scale Heat Sinks for High Heat Dissipation Rate Applications
,” Fluid Dynamics, e-print
arXiv:2106.11752
.10.48550/arXiv.2106.11752
17.
Jischa
,
M.
, and
Rieke
,
H. B.
,
1979
, “
About the Prediction of Turbulent Prandtl and Schmidt Numbers from Modeled Transport Equations
,”
Int. J. Heat Mass Transfer
,
22
(
11
), pp.
1547
1555
.10.1016/0017-9310(79)90134-0