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Flow of shear-thinning fluids in a concentric annulus

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Abstract

Distributions of mean axial velocity, axial and tangential turbulence intensities together with friction factor versus Reynolds number (f-Re) data are presented for three non-Newtonian liquids in fully developed laminar, transitional and turbulent flow in an annular geometry in the absence of centrebody rotation. Each of the non-Newtonian fluids was shear thinning and to some extent elastic and one was also thixotropic in character. For comparison purposes, measurements are also reported for a Newtonian fluid.

In the case of the Newtonian fluid, a mixture of glucose syrup and water, the f-Re data in both laminar and turbulent flow follow the appropriate relationships for the annular geometry, with a clear demarcation at transition which is confirmed independently by a measured increase in the centre-channel axial turbulence intensities. The measured velocity profiles for laminar flow are in good agreement with those predicted theoretically, whilst the turbulent profiles obey the log-law relationship over much of the mid-channel region and tend to the u +=y + relationship in the immediate vicinity of both walls.

For the first non-Newtonian fluid, an aqueous solution of sodium carboxymethylcellulose (CMC), good agreement with theoretical predictions for a power-law fluid was observed in the f-Re data in the laminar regime with evidence of drag reduction in turbulent flow. Velocity profiles, determined in two planes, indicate minor circumferential asymmetry in laminar flow. Law-of-the-wall plots for fully turbulent flow indicate an upward shift in the data in the log-law region of the annulus consistent with the drag-reduction behaviour, as also observed in pipe-flow experiments for this fluid (Escudier et al. 1992). In the near-surface regions of both the outer and inner tubes the data again tend towards the u +=y + relationship.

Anomalous behaviour was observed in the f-Re curves for the second non-Newtonian fluid, 0.125% and 0.2% aqueous solutions of Xanthan gum, with data for both concentrations falling significantly below the appropriate f-Re relationship for a power-law fluid. The anomalies are attributed to the elastic character of Xanthan gum. In the near-surface region of the outer tube the velocity-profile data again tend towards the u +=y + relationship but it proved impossible to obtain data in the near vicinity of the inner wall due to slight turbidity of the fluid.

The third non-Newtonian fluid, a Laponite/CMC blend, again exhibits anomalous f-Re behaviour, attributed to the thixotropic nature of this fluid. Velocity profiles determined in two planes again indicate some circumferential asymmetry in the laminar regime. Law-of-the-wall plots for the transitional and turbulent profiles tend towards the u +=y + relationship in both near-wall regions, again with an upward shift in the core of the annulus, consistent with drag reduction.

In general terms, the experimental results are consistent with previous work for non-Newtonian fluid flow in circular pipes and with limited data for an annular geometry (Nouri et al. 1993), with regard to drag reduction, modified turbulence structure and scale effects.

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Abbreviations

D i :

centrebody diameter (m)

D o :

outer pipe diameter (m)

(D o -D i ):

hydraulic diameter (m)

f :

friction factor (2 · τ A /ϱU2)

n :

power-law exponent (-)

p :

fluid static pressure (Pa)

Q :

volumetric flow rate (m3/s)

r :

radial distance from pipe centreline (m)

R i :

centrebody radius (m)

R o :

outer pipe radius (m)

R n :

refractive index (-)

Re :

Reynolds number ϱU(D o -D i )/μ s

s :

geometric scaling factor (-)

u :

mean axial velocity (m/s)

u′ :

rms fluctuation in axial velocity (m/s)

u′ c1 :

rms fluctuation in centreline axial velocity (m/s)

u :

non-dimensional value of u (u/u τ)

u τ :

friction velocity (m/s)

w′ :

rms fluctuation in tangential velocity (m/s)

x :

axial distance along pipe (m)

y :

distance from pipe or centrebody wall (m)

y + :

non-dimensional value of y (u τ y/v s )

Δp/L :

pressure drop per unit length (N/m2/m)

\(\dot \gamma\) :

shear rate (s-1)

κ :

radius ratio R i /R o

λ C :

constant in Cross model (s)

λ CA :

constant in Carreau model (s)

λ HB :

constant in Herschel-Bulkley model (s)

λ n :

constant in power-law model (s)

λ S :

constant in Sisko model (s)

μ :

dynamic viscosity (Pa · s)

μ ref :

reference viscosity (1 Pa · s)

μ s :

viscosity at wall at prevailing surface shear stress (Pa · s)

μ 0 :

zero shear-rate viscosity (Pa · s)

μ ∞ :

infinite shear-rate viscosity (Pa · s)

v :

kinematic viscosity (μ/ϱ) (m2/s)

v s :

kinematic viscosity at wall (m2/s)

ξ:

non-dimensional radial location (R o -r)/(R o -R i )

ϱ:

fluid density (kg/m3)

τ:

shear stress (Pa)

τ A :

weighted average wall shear stress (Pa)

τ i :

shear stress on centrebody (Pa)

τ o :

shear stress on outer wall (Pa)

τ s :

surface shear stress (Pa)

τ ref :

reference shear stress (1 Pa)

τ y :

fluid yield stress (Pa)

φ * :

geometry function of Jones and Leung (1981)

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Liverpool, L69 3BX, Liverpool, UK

    M. P. Escudier, I. W. Gouldson & D. M. Jones

Authors
  1. M. P. Escudier
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  2. I. W. Gouldson
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  3. D. M. Jones
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Additional information

The work reported here represents part of programme of research which has received financial support from SERC (GR/F 87813), BP Exploration Company Ltd, Shell Research BV and AEA Petroleum Services. This support is gratefully acknowledged. Frequent meetings with Professor J. H. Whitelaw, Imperial College of Science, Technology and Medicine, Dr. C. F. Lockyear and Dr. D. Ryan, BP Research, Ms. B. Kampman, Shell Research BV, and Dr. W. J. Worraker, AEA Technology, were of considerable benefit to the research.

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Escudier, M.P., Gouldson, I.W. & Jones, D.M. Flow of shear-thinning fluids in a concentric annulus. Experiments in Fluids 18, 225–238 (1995). https://doi.org/10.1007/BF00195092

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