Steady non–Newtonian flow past a circular cylinder: a ...

Acta Mechanica 172, 1?16 (2004) DOI 10.1007/s00707-004-0154-6

Acta Mechanica

Printed in Austria

Steady non?Newtonian flow past a circular cylinder: a numerical study

R. P. Chhabra, Kanpur, India, and A. A. Soares and J. M. Ferreira, Vila Real, Portugal

Received May 19, 2004; revised June 7, 2004 Published online: August 30, 2004 ? Springer-Verlag 2004

Summary. The steady and incompressible flow of non?Newtonian fluids past a circular cylinder is investigated for power law indices n between 0.2 and 1.4, blockage ratios of 0.037, 0.082 and 0.164, and the Reynolds numbers Re of 1, 20 and 40, using a stream function/vorticity formulation. The governing field equations have been solved by using a second-order accurate finite difference method to determine the drag coefficient, wake length, separation angle and flow patterns, and to investigate their dependence on power law index, blockage ratio and Reynolds number. The results reported here provide fundamental knowledge on the dependence of engineering flow parameters on blockage ratio and power law index, and further show that the effects on stream line and iso-vorticity patterns which result from an increase in the blockage ratio are similar to those which result from a decrease in the power law index.

1 Introduction

The flow of fluids past a circular cylinder represents a classical problem in fluid mechanics and thus has received considerable attention in the literature. Thus, over the years, the flow of a Newtonian fluid around circular cylinders has attracted a great deal of interest from the experimental, analytical, and numerical point of view, e.g. [1]?[5] and references therein, and indeed excellent reviews are available (e.g., [6]?[8]). An examination of these survey articles shows that an adequate body of knowledge (especially on the prediction of gross engineering parameters such as drag, wake characteristics, etc.) is now available on many aspects relating to the flow of Newtonian fluids past a long circular cylinder under most conditions of interest, albeit there are still some unresolved issues related to the detailed structure of the flow field. It is readily acknowledged that many materials encountered in industrial practice show non? Newtonian behavior [9]. Typical examples include polymer melts and their solutions, multiphase mixtures (suspensions, emulsions, foams) and soap solutions, etc. A knowledge of the hydrodynamic forces experienced by submerged objects like spheres and cylinders is needed in connection with the design of support structures and piers exposed to non?Newtonian muds, in the use of wires and thin cylinders as measurement probes and sensors in non?Newtonian flows and in the design of slurry pipelines where large particles are conveyed in a non?Newtonian vehicle. Additional examples are found in polymer processing operations such as the use of submerged surfaces to form weld lines. In addition to these potential applications, there is an intrinsic theoretical interest in elucidating the role of non?Newtonian characteristics on the structure of the flow field and on the drag for such highly idealized shapes like spheres and cylinders. In spite of such overwhelming importance and frequent occurrence of non?Newtonian fluids, there has been very little work reported on the non?Newtonian flow past a cylinder, even

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R. P. Chhabra et al.

for the simplest and possibly also the most common type of non?Newtonian behavior, namely, shear thinning and shear thickening behavior. This type of fluid behavior is frequently modelled using the simple two constant power law model. This work sets out to investigate the steady two-dimensional flow of power law liquids past a circular cylinder. It is, however, instructive to briefly summarize the previous scant results available in the literature prior to undertaking the formulation of the present problem.

From a theoretical point of view there have been only very limited studies reported in the literature. Thus, Tanner [10] demonstrated that in the limit of the zero Reynolds number there is no Stokes paradox for shear thinning fluids whereas it was still relevant for shear thickening fluids. This is clearly due to the fact that the effective shear rate in the fluid decreases rapidly away from the cylinder. For shear thinning fluids, this implies a progressive increase in the viscous forces whereas the viscous forces diminish for a shear thickening fluid. Subsequently a similar inference was also drawn by others [11]. Tanner [10] was able to obtain approximate analytical results for the creeping flow of power law fluids past a circular cylinder for shear thinning fluids. He also supplemented these results with numerical predictions in the range n ? 0:4 ? ?0:9; the correspondence was found to be reasonable. Subsequently, these results have been extended to even smaller values of the power law index up to n ? 0:2 [12]. However, all these results only relate to the zero Reynolds number and shear thinning fluid behavior. D'Alessio and Pascal [13] numerically investigated the steady power-law flow around a cylinder at Reynolds numbers Re ? 5, 20 and 40 using a first-order accurate difference method for a fixed blockage ratio b = 0.037, where b is defined as the ratio between the cylinder diameter and the distance H from the external boundary to the cylinder surface (Fig. 1). They investigated the dependence of critical Reynolds number, wake length, separation angle and drag coefficient on the power-law index. They reported that as the value of the Reynolds number was progressively increased there was a decreasing degree of convergence which restricted the range of power law indices for which a numerical solution was possible. Thus, for instance, for Re ? 20, they were able to obtain fully convergent results only for weakly non?Newtonian fluid behavior. Furthermore, their expression for the calculation of drag appears to be in error due to an inadvertent omission of a factor of 2. Fortunately, this does not influence their other results on wake characteristics, flow patterns, etc. However, it is also important to determine the dependence of the aforementioned parameters on the distance from the cylinder surface to the external numerical boundary (the blockage effect), since an increase in this distance approximates the conditions of flow in an infinite extent of fluid or, equivalently, decreases the wall

y q

field equations

H

upstream a

cylinder

r = ee

q downstream x

p

upstream

transformed field equations

cylinder

0

downstream

b

Fig. 1. The real (x; y) plane and the computational (e; h) plane

e? e

Steady non?Newtonian flow past a circular cylinder

3

effects. Although the effect of blockage on flow parameters and/or stream line patterns is well documented for Newtonian fluids (e.g., [14]?[16]) as well as for non?Newtonian viscoelastic fluids, both numerically and experimentally in the creeping flow region (e.g., [17]?[18]), a corresponding investigation for non?Newtonian viscoinelastic power law fluids is lacking. The aim of the present study is to extend the work of D'Alessio and Pascal [13] using a more accurate second-order finite difference method, more refined computational meshes and a greater power law index range in order to investigate the effect of blockage on drag coefficient, wake length, separation angle, and flow patterns (stream line and iso-vorticity contours) over wide ranges of conditions.

2 Basic theory

From a theoretical viewpoint, the steady incompressible flow of a power law liquid past a long

circular cylinder is described by the continuity and momentum equations. The oncoming steady

incompressible flow is in the x-direction normal to the axis of the cylinder. Due to the infinite

extent of the cylinder axis along the z-direction, the flow is two-dimensional, i.e., no flow

variable depends upon the z-coordinate and mz ? 0. The equation of continuity and the r and h

components of the equations of motion in cylindrical coordinates [19] can be expressed in terms

of the polar coordinates (e; h) with e ? ln?r=a?, giving the equation of continuity

1@ ee @e

ee @w @h

@ ?

@h

@w w?

@e

? 0;

?1?

the e-component

@w @h

@2w @e@h

?

@w @e

?

@2w w @h2

?

@w @e

?

w

?

?

1 2

@p @e

?

2n Re

!

e?e

@ @e

?ee

srr

?

?

@srh @e

?

shh

;

?2:1?

and the h-component

@w @2w

@w

@w

@2w

1 @p 2n

? @h

@e2 ? @e

?

?w @e

@e@h ? ? 2 @h ? Re

e?2e

@ @e

?e2esrh

?

?

! @shh ; @h

?2:2?

where the scaled dimensionless stream function w, vorticity x and pressure p are related to their dimensional counterparts as eewUa, e?exU=a and pU2q 2, respectively. U is the flow velocity,

a the stress

cylinder radius and q the tensor are related to their

fluid density. The dimensionless dimensional counterpart through

cKo?mUap?onsniej,ntasndsijthoef

the extra Reynolds

number Re is defined as

q?2a?nU2?n

Re ?

;

?3?

K

where K denotes the power law consistency index and n the power law index. For a shear thinning fluid n < 1, and n > 1 describes shear thickening behavior.

The constitutive equation for a power law fluid is written as

sij ? ?geij;

?4?

where g is the dimensionless viscosity and eij are the dimensionless components of the rate-ofdeformation tensor (e.g., [19]).

The equation for the power law dimensionless viscosity is

n?1

g ? I22 ;

?5?

4

R. P. Chhabra et al.

and I2 is the dimensionless second invariant of the rate-of-deformation tensor given as

I2

?

" e?2e J2

?

@2w 2#

4

;

@h@e

?6?

with

@2w @2w

J ? w ? @e2 ? @h2 :

?7?

The vorticity in its scaled form is given as

@2w @2w @w

@e2 ? @h2 ? 2 @e ? w ? x ? 0:

?8?

Eliminating the pressure in Eqs. (2) by the method of cross-differentiation and introducing the

vorticity x with some rearrangement leads to

@2x

@

2

x

@x

@x

g @e2 ? @h2

? 2k ? 2l ? cx ? F;

@e

@h

?9?

where

@g

Re ee @w

k ? @e ? g ? 2n?1 @h ;

?10?

@g

Re

ee

@

w

l ? @h ? 2n?1 @e ? w ;

?11?

@g

Re ee @w

c ? ?2 ? g ? @e

2n

; @h

?12?

F

?

@

2g

J @h2

?

@2g @e2

?

2

@g @e

?

4

@2w @h @e

@g @h

?

@2g :

@h @e

?13?

Due to the two-dimensional nature of the problem (xy-plane) and since the oncoming flow is in the x-direction, we only need to consider the region y ! 0 and x2 ? y2 ! 1. Thus, the corresponding region in the (e; h)-plane is defined by e ! 0 and 0 h p (Fig.1).

The boundary conditions are expressed as follows: On the cylinder surface, the usual no-slip condition is applied, i.e.,

@w ? @w ? 0 for e ? 0: @e @h

?14:1?

Equation (14.1) together with Eq. (8) gives

@2w w ? 0 and x ? ? @e2 :

?14:2?

Along the x-direction,

w ? x ? 0 for h ? 0; p:

?14:3?

Since both the stream function and vorticity equations are of elliptic nature, it is necessary to establish boundary conditions for w and x at the external boundary, i.e., at a large distance (r1) from the cylinder. Imai [20] has given asymptotic formulas for the stream function and vorticity which are generally applied as external boundary conditions for both Newtonian (e.g., [3], [4], [14] ) and power law [13] fluids. In polar coordinates, the first terms of w and x at the external boundary are obtained inserting the viscosity equation (5) into Imai's [20] equations, giving the two following external boundary conditions (both in scaled and dimensionless form):

Steady non?Newtonian flow past a circular cylinder

5

w % sin?h? ? Cd e?e h ? erf?Q? ;

?15?

2p

1?n

x % ? C2dn?R1epI2ffipffi2ffi Qe?Q2 ;

?16?

where Cd is the drag coefficient,

rffiffiffiffiffi

Q

?

e

e2

Re 1?n

h

2n I24 sin 2 ;

?17?

and erf(Q) is the standard error function. It is appropriate to mention here that for n ? 1,

Eqs. (16) and (17) reduce to the Newtonian situation (e.g., [3], [4], [14]). Although Eqs. (16) and

(17) differ from those of D'Alessio and Pascal [13] by numerical factors of 0.5 and 1.4,

respectively, their corresponding equations appear to be in error, as they do not reduce to the

expected Newtonian limiting behavior for n ? 1.

The exponential scaling for the stream function and vorticity is appropriate since the stream

function is exponentially large far from the cylinder, and the vorticity is exponentially small

everywhere except in the region of the wake [13]. For the external boundary conditions used in

the present study, the scaling procedure keeps the values of w between zero and 1 and x

between zero and )4, and thus suppresses numerical instabilities.

Once the values of x, w and g are known in the flow domain, the total drag coefficient is given by

2n?1 Zp Cd ?

Re

"I2n?2 1

@x" @e

?

x"

?

n

? 2

1

@"I2 @e

"I2n?2 3

!

x "

e?0

sin?h?dh;

0

?18?

where x" is the dimensionless vorticity in real space (x" ? e?ex), and "I2 is the dimensionless

second invariant of the rate-of-deformation tensor in real space which is obtained through the insertion of the dimensionless stream function in real space, w ? e?ew", into Eq. (6). Finally, it is

also appropriate to add here that the factor of 2 in the denominator of the rightmost term in

Eq. (18) is missing from the corresponding equation in [13].

3 Numerical solution method

The numerical solutions obtained for the computational domain shown are in Fig. 1b. For an (N+1) ? (M+1) computational mesh, the spacings in the e- and h-directions are e1=N and p=M, respectively. The governing stream function and vorticity equations (8) and (9) are rewritten as finite difference equations using the central difference of second-order accuracy. A second-order upwind differencing technique [21] is used to solve Eq. (9) with one-sided difference approximations to the first derivatives of x. Numerically, this technique yields a tridiagonal matrix which is diagonally dominant and, therefore, unconditionally stable. The second order upwind differencing technique was used in the present study in preference to the central difference approximation, because preliminary tests showed that the latter approximation resulted in numerical instability, consistent with observations by other authors (e.g., [22]). The steady-state solutions for the governing equations, namely stream function (Eq. (8)), vorticity (Eq. (9)) and power law viscosity (Eq. (5)), are obtained using the Gauss-Seidel relaxation iterative method [23]. To obtain consistent approximations for all variables, for each iteration a sweep is made through all mesh points and an updated value of the drag coefficient is determined by numerical integration of Eq. (18) on the cylinder surface using Simpson's rule.

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