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A computational model of wall shear and residence time of particles conveyed by steady flow in a curved tube
Marc Thiriet, J. Graham, R. Issa
To cite this version:
Marc Thiriet, J. Graham, R. Issa. A computational model of wall shear and residence time of particles conveyed by steady flow in a curved tube. Journal de Physique III, EDP Sciences, 1993, 3 (1), pp.85- 103. �10.1051/jp3:1993122�. �jpa-00248910�
Classification Physics Abstracts
02.60 47.60 87.45
A computational model of wall shear and residence time of
particles conveyed by steady flow in a curved tube
M. Thiriet (1.2), J. M. R. Graham (2) and R. I. Issa (2)
(~) L-B-H-P- URA 343 CNRS, Universitd Paris VII, 2 Place Jussieu, 75251Paris Cedex05,
France
(2) PFSU and Department of Aeronautics, Imperial College, Royal School of Mines, Lon- don SW 7, Great Britain
(Receii,ed 3 April 1991, revised 5 March 1992, accepted ii September 1992)
Rdsumk. Les dquations de conservation de la masse et de la quantitd de mouvement ont dtd rdsolues pour un dcoulement stationnaire d'un fluide incompressible visqueux dans un coude
(angle de 90°), de parois lisses et rigides et de section droite uniforme et circulaire, par la mdthode des volumes finis. Les conditions limites en entrde pour l'dcoulement laminaire sont soit un profil
parabolique, soit un profil uniforme de vitesse axiale, pour des valeurs du nombre de
Dean140w De w430. Un calcul numdrique de l'dcoulement turbulent a dtd effectud pour
De 22 loo avec un profil de vitesse plat en entrde du mod61e. La vitesse axiale ddpend non
seulement de la valeur du nombre de Dean, mais aussi du nombre de Reynolds et de la courbure du tube sdpardment. Le profil de vitesses est perturbd pour de trks faibles courbures. Le trongon de conduit oh les mouvements secondaires sont les plus forts est situd d'autant plus prks de l'entrde du coude que la couche limite en entrde est plus dpaisse. L'dcoulement secondaire se ddveloppe sur
une rdgion plus dtendue du coude et le cisaillement paridtal est plus dlevd quand le nombre de Reynolds augmente. Le coude affecte le cisaillement paridtal en amont et en aval. La prdsence d'un coude dans une conduite fait croitre le temps de rdsidence des particules en suspension par rapport
aux valeurs prises dans un tube droit isold uniquement dans l'dldment de conduite droit situd en aval du coude. Quand le nombre de Reynolds augmente, le rdgime d'dcoulement restart laminaire,
cette caractdristique disparait. Elle est cependant observde en dcoulement turbulent, le temps de
rdsidence prenant en tout point des valeurs beaucoup plus faibles que celles obtenues en rdgime
laminaire.
Abstract. A finite-volume model of steady flow of an incompressible viscous fluid has been carried out in a smooth rigid 90° bend of circular cross-section. The inlet boundary conditions for laminar flow are either an entry Poiseuille regime or a constant injection velocity for a range of Dean number 140 w De w 430. A numerical test of turbulent flow was performed for De
,
22 loo with a flat velocity profile at the model entry. The lower the role played by the viscous forces, the
larger the distance necessary to set an outer shift of the peak axial velocity. The axial velocity of laminar flow depends not only on the value of the Dean number, but also on separate effects of the Reynolds number and of the tube curvature. The larger the laminar boundary layer at the bend inlet, the nearer from the entry the bend segment where the strongest secondary motion is located.
With increasing Reynolds number, the secondary flow develops over a longer bend region and the wall shear rises. Upstream and downstream effect of the bend on the shear stress, as well as flow
disturbances induced by very small curvature, were observed. The residence time of conveyed
particles is enhanced by the presence of a curved section in the conduit with respect to a straight pipe only at the inner edge of the straight section located downstream from the bend. When the Reynolds number rises, the flow regime remaining laminar, the residence time is smaller in the whole pipe. However for turbulent flow, the residence time, which has much smaller values, takes its highest values in the exit straight section.
Nomenclature
a tube radius
d tube diameter
De Dean number
k curvature ratio
K kinetic energy of turbulence
p pressure
R radius of curvature
Re Reynolds number
t time
u circumferential velocity
v radial velocity
w axial velocity
x circumferential angular coordinate
y radial coordinate
z axial (angular in bend) coordinate
3 boundary layer thickness
e rate of dissipation of turbulent kinetic energy
m fluid viscosity
p fluid density
~ Prandtl number
T shear stress
SUBSCRIPT.
a axial
c circumferential
I laminar
t turbulent
r residence
Introduction.
In curved tubes, centrifugal forces induces an helical motion, as shown by flow visualisation
experiments carried out by Eustice in 1911. Velocity vectors can be decomposed in a stream- Wise component w and a component at right angle to the tube axis tt + v, the so-called secondary flow velocity. In the cross-section, the fluid along the plane of curvature moves
toward the outside of the bend and the fluid near the Wall flows circumferentially back to the inner Wall. The spiral motion promotes the lateral mixing of the conveyed particles. In a short
entrance region of the bend, the fluid velocity is higher near the inner wall due to the lateral
pressure gradient. Further downstream, the peak axial velocity is located near the outside
edge [23, 2]. The distortion of the flow pattem caused by the tube curvature is associated with
an increase in resistance. Dean (1927) showed theoretically on fully developed laminar flow, for small curvature, the dependence of the resistance on a dimensionless quantity, the so-called
Dean number (De
= k~~~ Re, where k is the curvature ratio, k
= a/R, a tube hydraulic radius, R radius of curvature of the tube axis, and Re the Reynolds number based on the tube diameter
and on the mean axial velocity). This dependence was confirmed by the experimental work of White (1929).
The direction of the single vortex system in each cross-section, which was first demonstrated
by Dean (1927), can be explained after breaking up the inlet flow into an irrotational component and a vortex ring. The circulation direction of the vortex ring goes from the « core » to the « boundary layer » in the streamwise direction (clockwise rotation with respect to the vorticity direction). This vortex ring is axially stretched by the curvature. In the downstream
segment of the 90° bend, the angular momentum at both its downstream and upstream ends (originally located at the inner and outer wall in the bend entrance region) weakens, whereas the circulation of the vortex ring at both upper and lower walls generates the single vortex
system in each half cross-section, directed toward the outer wall along the centerplane. The
magnitude of the secondary motion increases with Re and k [14]. Besides, the secondary flow
develops over a length scale of (aR )~~~, whether the entry axial velocity profile is flat [24, 14]
or parabolic [4]. When De rises, the complexity of the secondary motion increases, with a
counter-rotating eddy linked with the cross-flow reversal near the centerplane [10, 19, 16].
The flow in curved pipes is more stable than in straight pipes [21, 22, II. In turbulent flow, the outward shift of the axial velocity occurs only in the downstream part of a 90° bend [8]. The axial velocity profiles in a plane normal to the curvature plane at the tube axis remain nearly
constant for a wide range of De [15, 3].
The present work is aimed at studying developing flows in a 90° bend, both in laminar (140 « De « 430) and turbulent conditions. Because the curvature stabilizes the flow and
delays transition to turbulence, the turbulent test was run with a high De (De
=
22 loo). The conditions at the inlet of the straight tube upstream from the bend entry are either a uniform injection velocity, a developing flow or a fully-developed flow. The study is mainly focused on the spatial variations of the wall shear and the resulting residence time in order to determine the interaction between the flow behaviour and the particle deposition in curved pipes. Although physiological flow are unsteady (the respiratory flow is purely sinusoidal and the blood flow is
pulsatile) this work deals with steady flow. Numerous literature data on steady flow in curved
pipes are available, and can be used to validate the numerical model. However, complete sets of data on shear stress and residence time were lacking, especially the effect of the bend on the
ducts located upstream and downstream from it, as well as the role played by the tube
geometry and the goveming parameters. These results can be afterwards compared to the data of numerical experiments carried out on unsteady flow. Yet, the residence time of a conveyed particle can not be computed when the fluid moves back and forth from the flow velocity
components.
Method.
The fluid velocity u and the pressure p were solved using the classical equations of fluid mechanics
(I) mass conservation V u
=
0
(2) momentum conservation : p(u V) u m Au + Vp
= 0 ;
where p is the fluid density and m the fluid viscosity.
A finite-difference scheme is used to solve the developing flow of incompressible fluid
through a 90° bend. The curved section is interposed between 2 straight sections, a short
(length of 3.6 d) upstream and a longer one (length of 8. I d~ downstream. Briefly, the finite- volume method employs an hybrid of upwind and central differencing procedure proposed by
Caretto et al. (1972). The solution procedure used the PISO algorithm of Issa (1982). The sequence of main operations of the solution procedure is as follows : (I) the pressure field is
guessed, (it) the predictor-step momentum equations are solved to get a first approximation of the 3 components of the velocity field, (iii) the first corrector-step pressure-increment
equations is solved to obtain a new pressure field, which gives a corrected velocity field
satisfying the continuity equations, (iv) the second corrector-step pressure-increment equation
with new pressures calculates new velocities. The residuals for each equation solved over the entire field must have decayed systematically below a specified maximum tolerable value to obtain a convergence. Otherwise a new iteration with the same steps is initiated. The velocity
field in the 3 directions is computed using a line-by-line counterpart of Gauss-Seidel iteration
employing the W-diagonal matrix scheme. The pressure field is calculated by the Stone
implicit method [20]. The number of line iterations at each complete sweep through the domain varies for each variable ; eight iterations are used for the 3 components of the velocity and 250 for the pressure.
The « velocity nodes » are staggered with respect to the storage locations of all other variables. The domain is discretised into a finite number of hexahedra. The pressure is stored at the center of the cell while the three components of the velocity tt (circumferential direction x), v (radial direction y) and w (axial direction z) are defined at the center of the faces of the
hexahedron to which these velocities are normal (Fig. I). Due to the limitation in memory size
(using a CDC CY855 under NOS operating system), a relatively coarse mesh (7 x 7 x 48)
was used. Figure 2 shows the longitudinal mesh of the fluid domain which consists of a 90°
bend between 2 straight tubes.
Uii,J,~i)
O
1,J,K)
dxII J+I KJ
~'f II-I,J K)
' '
O '
(J,K)
'" Iii,K)
° + , s (I,~i,K)
o
K)
o dy II,)
dxii,»I)
ii,)) W(I,J,K+1) D II,),K+11
O
Fig. I. Main control volume (center P) and layout of computed variables. x is the circumferential coordinate (index of discretization I), y the radial coordinate (index J), and
z the longitudinal coordinate (index K). S, N, W, E, U, D are the adjoining nodes, where the pressure is stored.
32.4°
b~ pO
bj
Fig. 2. Grid system and monitoring stations along the pipe model. Dots at the cell center give the streamwise coordinate of nodes used to compute the pressure, the cross velocity and the shear stress.
Arrows indicate the cross-sections for axial velocity plots. bi bend inlet, bo : bend outlet, pi pipe inlet, po pipe outlet.
The simulation of turbulent flow is achieved by the use of the K
e model of turbulence
developed by Launder and Spalding (1972), assuming that the turbulence induces diffusion- like processes. The turbulent viscosity is determined from the time-mean values of the kinetic energy of turbulence K, and the volumetric rate of dissipation of turbulent kinetic energy
e by
JL~ = C~ pK~le (C~
= 0.09).
The model contains 4 other empirical constants, C
i = 1.44, C~
= 1.92, the turbulent Prandtl
number for the diffusive transport of K and e, ~k " 0 and
~~ = w ~/((C~ C
i C)~ = l.22 (w
=
0.4187 ). The « effective » coefficients of exchange
become JL
= JLt + JL~ (where the subscript I refers to laminar quantity) for the 3 velocity components, JL/~~ and JL/~~ for K and e respectively. The distinction between the near-wall nodes located in the turbulent region and those lying in the laminar sublayer is made by the value of a variable y~ defined by Launder and Spalding (1972) as y~
= (pC)~K~~~/mt y]~.
When y~ ~ l1.63, the point P is located in the laminar sublayer and the wall shear stress T~ is given by T~= m~W~/3 where 3~ is the thickness of the layer. Otherwise, T~ is expressed as T~
=
[pC(~ w~~~W/In (Ey~ )]~, where E
=
9.793 for a smooth wall.
Furthermore, the source terms of the finite-difference form of the goveming differential
equation for K and e, at nodes adjacent to the wall, are modified accordingly.
After initialisation of the working arrays, the input values and the grid information are get and the radii of curvature are computed. In the main loop, a call is made to subroutines which
solve the finite-difference momentum equations in the 3 directions, taking into account
updated boundary values. A call to a subroutine which solves the pressure equation follows and then 3 subroutines solves eventually the temperature, the kinetic energy of turbulence and the energy-dissipation rate. The number of outer iterations, corresponding to the main loop,
varied between 70 and 100 with the present flow conditions.
Results.
THE VELOCITY FIELD COMPARISON WITH LITERATURE DATA AND EFFECTS OF THE FLOW
PARAMETERS. Figure 3 shows the results of an entrance Poiseuille flow test, compared to
the complete set of data provided by Bovendeerd et al. (1987), for k
= 1/16, Re
=
700, De
= 286. Axial velocity profiles in the plane of curvature for every section of measurement
within the bend (at 0.24 d, 0.61d, 1.23 d, 2.08 d, 3.06d and 4.29 d) obtained by these
investigators are compared with the numerical profiles at the closest sites (I.e. at 0.19 d, 0.57 d, 1.13 d, 2.07 d, 3.02 d and 4.33 d~. The outward shift of the peak velocity occurs earlier in the computational model. Then the velocity maximum is lower than the experimental value, and the difference is greater after 50° of curvature. In the inner part of the symmetry plane, the
velocity plateau is longer and disappears later. The wall shear appears to be overestimated at the inside wall. In agreement with the results of Bovendeerd et al. (1987), a slight inward shift of the axial velocities towards the inner wall is observed in the bend entrance region. Besides, the velocity profiles at two neighbouring stations (43.2° and 46.8°) are quite different.
Furthermore, the axial velocity profile are very similar in the downstream part of the bend.
But, the existence of slight differences states that there is no fully-developed flow regime
before 90° of curvature.
The developing axial velocity field is also represented in figure 4, by isovelocity contours.
The contour plot uses as approximant a seamed quadratic finite element which achieves a high
degree of accuracy (Sibson and Thomson, 1981). The concentric contours are only slightly
distorded in the first 15° of curvature. A shift of the contours towards the outer wall appears then clearly ; the region of lower velocity spreads out towards the center and the upper wall. In
_i _i _i
L
c
~
c
-i -i i -1
Fig. 3. Comparison between the numerical values (C) and the laser-Doppler velocity measurements
(L) of Bovendeerd et al. (1987), at about 4°, 11°, 22°, 40°, 58° and 82° from top left to bottom right. The inner wall ~y =
I is at left.
agreement with the findings of Bovendeerd et al. (1987), a region of velocity lower than 0.8 is observed along the symmetry plane in the inner half of the bend, in the distal part of the curved tube. The axial shear is much greater in the outer half of the bend wall. The peak velocity is
o
~ ;. 75.6
'e'
j/'>I
j
""
a '. 4«, ,,,
/
f
/.. ' ,,~j 14.4
~, ~
86.4
,_=~~-
l ~
' 'i ,"
~~ ~~
32 4 ' ,~ "' l.7d
.'
~ fi)(j j)
(~ l)ii~lj ' ii '
/ /,' j ' (
'
/ ,' 'I
<' ~ / ~ 'il,M' / ''"~
~ /fj,,/
>'>
~$
-~ W~
' / ~,
/
/ b.<
, ;
"fl'
~~ ~ " 3 6 d
_J
~
°
61.2
~
Fig. 4. Contours in the half cross-section of the non-dimensional axial velocity W/W (W mean
velocity) at the bend inlet, at14.4°, 32.4°, 48.6°, 61.2°, 75.6°, 86.4°, +1.7d and +3.6d (k = 1/6, Re
=
700, De
=
286). 0
= outer wall, I
= inner wall.
displaced from the plane of curvature ; consequently a double peak can be observed in the entire tube cross-section. Such a feature has also been observed by Enayet et al. (1982), with
laminar flat entrance flow at relatively high Re.
Although a « parabolic-like » velocity profile persists in the bend inlet, a secondary flow is
already generated. The secondary motion increases greatly and seems to be stronger between approximately 15 and 50° of curvature. The circumferential back flow occurs through a thin
layer along the upper wall with a maximum at the top edge. At 34°, the secondary velocity is greater in the outer bend, while at 63°, higher secondary velocities are observed at inner bend.
These results are similar to those of Bovendeerd et al. (1987). The secondary flow decreases then, while the swirl center is moving. In the downstream straight pipe, the secondary motion
decreases sharply at + 2 d, it is lower than at the bend inlet. However the secondary flow intensity might be slightly reduced by the exit boundary condition. The computation reveals that the secondary resultant component is 20 fb of the axial component, when De = 279, in the first part of the curved tube, more especially near 31°, I-e- in the region located at
,
1.7 (aR)~~~ with R
= lo a, where the secondary flow is maximum [4]. Besides, the ratio is 6 fb, + 2 d downstream from the bend exit.
Laser-Doppler measurements of the developing laminar flow have also been carried out by Agrawal et al. (1978) in a 180° bend (k
= 1/7 and 1/20, 138 « De « 679) directly connected to an upstream reservoir via a convergent nozzle. Numerical simulations of this flow have been
L
c c
~
-i i -i o i -i
1.5j
l(
L
/v/
o.5-(1' '~
>/
o ~
-I I -I o I -I o
Fig. 5. Development of the axial flow in a 90° bend (k
= 1/7. De
= 183). Steady laminar flow,
constant injection velocity at the bend entry. Comparison with literature data. Top laser-Doppler
measurements (L) of Agrawal et al. (1978) in a 180° bend at 15.1°, 30. I and 60.I from the bend entry
(selected cross-sections of the computational model (C) at 14.1°, 28.8°, 61.2°). Bottom finite-difference model (L) of Soh and Berger (1984) corresponding to the physical model of Agrawal et al. (1978), with
constant dynamic pressure as inlet condition. From left to right, at 60° (C at 61.2°), 75° (C at 75.6°), 83°
(C at 82.8°).
