In this paper, an analytical solution is found for the Reynolds equations describing a simple turbulent shear flow carrying small, wake-less particles. An algebraic stress model is used as the basis of the model, the particles leading to source terms in the equations for the turbulent stresses in the flow. The sources are proportional to the mass loading of the particles and depend on the temporal correlations of the fluid velocities seen by particles, Rijτ. The resulting set of equations is a system of nonlinear algebraic equations for the Reynolds stresses and the dissipation. The system is solved exactly and the influence of the particles can be quantified. The predictions are compared with DNS results and are shown to predict trends quite well. Different scenarios are investigated, including the effects of isotropic, anisotropic and non-equilibrium time scales and negative loops in Rijτ. The general trend is to increase anisotropy and attenuate turbulence with higher mass loadings. The occurrence of turbulence enhancement is investigated and shown to be theoretically possible, but physically unlikely.

Issue Section:
Additional Technical Papers
Keywords:
shear turbulence, nonlinear equations, multiphase flow, algebra
Topics:
Anisotropy, Particulate matter, Stress, Turbulence, Shear turbulence, Flow (Dynamics), Algebra
1.
Gore
,
R. A.
, and
Crowe
,
C.
,
1989
, “
Effect of Particle Size on Modulating Turbulent Intensity
,”
Int. J. Multiphase Flow
,
15
(
2
),
279
–
285
.
2.
Hetsroni
,
G.
,
1989
, “
Particle-Turbulence Interaction
,”
Int. J. Multiphase Flow
,
15
(
5
),
735
–
746
.
3.
Savolainen, K., and Karvinen, R., 1998, “The effect of Particles on Gas Turbulence in a Vertical Upward Pipe Flow,” Third International Conference on Multiphase Flow, ICMF’98, Lyon, France.
4.
Varaksin, A. Y., Kurosaki, Y., Satoh, I., Polezhaev, Y. V., and Polyakov, A. F., 1998, “Experimental Study of the Direct Influence of the Small Particles on the Carrier Air Turbulence Intensity for Pipe Flow,” Third International Conference on Multiphase Flow, ICMF’98, Lyon, France.
5.
Crowe
,
C. T.
,
1999
, “
On Models for Turbulence Modulation in Fluid-Particle Flows
,”
Int. J. Multiphase Flow
,
26
(
5
),
719
–
727
.
6.
Berlemont
,
A.
,
Desjonqueres
,
P.
, and
Gouesbet
,
G.
,
1990
, “
Particle Lagrangian Simulation in Turbulent Flows
,”
Int. J. Multiphase Flow
,
16
(
1
),
19
–
34
.
7.
Eaton, J. K., 1995, “Turbulence Modification by Particles in Shear Flows,” paper presented at the Symposium on Gas/Particle Flows, ASME FED Summer Meeting, Hilton Head, SC, USA.
8.
Squires
,
K. D.
, and
Eaton
,
J. K.
,
1990
, “
Particle Response and Turbulence Modification in Isotropic Turbulence
,”
Phys. Fluids A
,
2
,
1191
–
1203
.
9.
Graham
,
D. I.
,
2000
, “
Turbulence Attenuation by Small Particles in Simple Shear Flows
,”
ASME J. Fluids Eng.
,
122
(
1
),
134
–
137
.
10.
Libby, P. A., 1996, “Introduction to Turbulence,” Taylor and Francis, New York, USA.
11.
Graham, D. I., 1998, “Turbulence Modulation by Particles in Homogeneous Shear,” paper 5008, ASME FED Summer Meeting, Washington DC.
12.
Tavoularies
,
S.
, and
Karnik
,
U.
,
1989
, “
Further Experiments on the Evolution of Turbulent Stresses and Scales in Uniformly Sheared Turbulence
,”
J. Fluid Mech.
,
204
,
457
–
478
.
13.
Simonin, O., Deutsch, E., and Boivin, M., 1995, “Large Eddy Simulation and Second-Moment Closure Model of Particle Fluctuating Motion in Two-Phase Turbulent Shear Flows,” in Selected Papers from the Ninth Symposium on Turbulent Shear Flows, eds F. Durst, N. Kasagi, B. E. Launder, F. W. Schmidt, and J. H. Whitelaw, Springer-Verlag, Germany.
14.
Taulbee
,
D. B.
,
Mashayek
,
F.
, and
Barre
,
C.
,
1999
, “
Simulation and Reynolds Stress Modelling of Particle-Laden Turbulent Shear Flows
,”
Int. J. Heat Mass Transfer
,
20
,
368
–
373
.
15.
Champagne
,
F. H.
,
Harris
,
V. G.
, and
Corrsin
,
S.
,
1970
, “
Experiments on Nearly Homogeneous Shear Flow
,”
J. Fluid Mech.
,
41
,
247
247
.
16.
Rose
,
W. G.
,
1970
, “
Interaction of Grid Turbulence With a Uniform Mean Shear
,”
J. Fluid Mech.
,
44
,
767
767
.
17.
Grosman
,
A. D.
, and
Ioannides
,
E.
,
1981
, “
Aspects of Computer Simulation of Liquid-Fueled Combustors
,” AIAA paper 81-0323, 19th Aerospace Science meeting, St. Louis, USA.
18.
Squires
,
K. D.
, and
Eaton
,
J. K.
,
1994
, “
Effect of Selective Modification of Turbulence on Two-Equation Models for Particle-Laden Turbulent Flows
,”
ASME J. Fluids Eng.
,
116
,
778
–
784
.
19.
Boivin
,
M.
,
Simonin
,
O.
, and
Squires
,
K. D.
,
1998
, “
Direct Numerical Simulation of Turbulence Modulation by Particles in Isotropic Turbulence
,”
J. Fluid Mech.
,
375
,
235
–
263
.
20.
Boivin, M., 1996, “Etude de l’Influence des Particules sur la Turbulence a partir de Simulations Directes et de Simulations des Grand Echelles d’Ecoulements Diphasiques Gaz-Solides Homogenes Isotropes Stationnaires,” Report HE-44/96/010/A, Departement Laboratoire National d’Hydraulique, Electricite de France, France.
21.
Graham
,
D. I.
,
1997
, “
Turbulence Modification in the Limiting Cases of Heavy- and Tracer-Particles
,”
ASME J. Fluids Eng.
,
119
(
2
),
458
–
460
.
22.
Riley
,
J. J.
, and
Corrsin
,
S.
,
1974
, “
The Relation of Turbulent Diffusivity to Lagrangian Velocity Statistics for the Simplest Shear Flow
,”
J. Geophys. Res.
,
79
(
2
),
1768
–
1771
.
Copyright © 2004
 by ASME
You do not currently have access to this content.