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Permeability for flow normal to a sparse array of fibres
ELSEVIER
Materials Science and Engineering
Al 91 (1995) 203-208
Permeability for flow normal to a sparse array of fibres D. Nagelhouta, M.S. Bhatb, J.C. Heinricha, D.R. Poirierb “Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 8.5721, USA bDepartment of Materials Science and Engineering, University ofArizona, Tucson, AZ 85721, USA Received 7 March 1994; in revised form 27 July 1994
Abstract Solutions for flow normal to an array of fibers (visualized as circular cylinders) arranged in a square array were determined using a finite element formulation. The calculated results were used to obtain the permeability for flow through the system with as little as 0.01 volume fraction solid and for a Reynolds number up to 40. These results can be used to estimate permeabilities for infiltrating fiber composites or for modeling the solidification of dendritic alloys with transport phenomena in porous media. Keywords: Fibres; Permeability
1. Introduction
p-Kvp I
Ever since fiber composites were produced by the liquid metallurgy route, their solidification behavior and microstructural development have been the subject of intense research. Mortensen et al. [l] used Darcy’s law to analyze the infiltration of metal through fibers for the production of metal matrix composites. Recently Sekhar and Trivedi [2] reported work on the interaction between fibers, narrow channels, and solidification fronts in a transparent analog system. Their study provides clear evidence for the strong effect of fiber arrangements on microsegregation patterns. It is also known that the convection of the interdendritic liquid through the dendritic mushy zone in solidifying alloys is responsible for most forms of macrosegregation in castings and ingots [3-51. The nature and extent of the segregation can be modeled as transport phenomena in porous media [6-81; therefore, to model the convection and its attendant macrosegregation in solidifying alloys, either the permeability as found in Darcy’s law or a drag coefficient derived from the permeability is required. According to Darcy’s law, the seepage velocity (also called the superficial velocity) U and the applied pressure gradient Vp are related by the expression [9], 0921-5093/95/$9.50 0 1995 - Elsevier Science S.A. All rights reserved SSD/O921-X)93(94)09641-4
I
11
where 11is the viscosity of interdendritic liquid and K is the permeability of the mushy zone. The permeability itself relates directly to the microstructural length scale of the solid comprising the dendritic network or fiber array and to the volume fraction of liquid. Utilizing these metrics, Darcy’s law can be rearranged to obtain the dimensionless permeability KS,’ 16U KS,’ = ~ VP
(2)
where S, is the specific surface area of the solid, U has been converted to the non-dimensional seepage velocity and Vp to the non-dimensional pressure gradient. Eq. (2) is derived in Appendix A. In most porous media, the fluid phase convects in a network of a uniform volume fraction of solid. However, in a solidifying alloy the interdendritic liquid moves in a solid network in which the volume fraction of liquid g, is not uniform. In the interior of the mushy zone (e.g. where 0 Ig,50.7), the momentum equation can be satisfactorily represented by Eq. ( 1). In the context of dendritic solidification, however, Eq. (1)
204
LI. Nagelhout et al.
/
Materials Science and Engineering A191 (199.5)203-208
gives erroneous results in the part of the mushy zone that is adjacent to the all-liquid zone, where g, exceeds 0.7, and it is necessary to use a comprehensive momentum equation with inertial terms and diffusion terms, along with the Darcy term. Also, the complete momentum equation should provide for a smooth transition of the momentum transport, from the mushy zone to the all liquid zone [S]. Specifically, as g,- 1, K -+ ~0 and the momentum equation reduces to the Navier-Stokes equation for a single-phase liquid. In order to characterize the permeabilities in dendritic solidification processes, it is necessary to consider flow in three categories: (1) flow through a columnar dendritic network with flow parallel to the primary dendrite arms; (2) flow through a columnar dendritic network with flow normal to the primary dendrite arms; (3) flow through a network of equiaxial grains. This paper addresses the second flow situation. There are no empirical data for permeability for flow through columnar dendritic networks when g,_> 0.66, and permeabilities cannot be measured at higher values of g, [lO,ll]. Therefore, we rely on direct numerical simulations of flow through dendritic microstructures and analytical results of flow through “model” microstructures. Sangani and Acrivos [ 121, Sparrow and Loefler [ 131, and Larson and Higdon [14] calculated the drag force for Stokes flow through two-dimensional periodic arrays of circular cylinders. Also, Launder and Massey [ 151, Thorn and Apelt [ 161 and Eidseth et al. [ 171 used numerical methods to calculate drag forces and permeabilities of flows with finite Reynolds numbers, but in these works the porosities were limited to a maximum of 0.5. More recently Edwards et al. [18] analyzed flows for Reynolds number up to 180, while extending the porosity to 0.8. For application to solidification scenarios, however, the volume fraction of liquid goes all the way to unity, and infiltrated fiber composites also contain less than 0.2 volume fraction of fibers. Hence, the present paper aims at calculating the permeability at a very high volume fraction of fluid, for which no empirical data are available. This is accomplished using a numerical method, based on finite elements, to calculate the two-dimensional flow normal to circular cylinders arranged in square lattices. By altering the ratio of diameter to the center-to-center distance between cylinders, the porosity can be made as high as 0.99.
> >
>
U Fig. 1. Flow through a square array of circular cylinders.
au
au
ap
ug+uaL'=-~+q
(3)
and
a% a% ,,a”+,atl_ap+ vg+ayz ax ay ay i i subject to the incompressibility
For the two-dimensional flow through the array shown in Fig. 1, the components of the Navier-Stokes equation are
constraint
au+!!%0
ax ay
where u and u are the x- and ycomponents of the velocity respectively, p is pressure, and q is the viscosity. A computational domain with three semicylinders is shown as Fig. 2. The cylinders have a radius R, and the center-to-center distance between them is 2L. Flow is symmetrical at the planes y= 0 and y= L. The boundary conditions for velocity are as follows: v =0
at
y= 0
v =0
at
y=L
and u = ug (uniform)
at
Along the solid-liquid sure we simply specify 2. Formulation
(4)
x=0 interfaces,
u =U = 0. For pres-
p( 0, L ) = p. (constant)
The numerical solutions are obtained in non-dimensional form using a finite element algorithm. In order to non-dimensionalize the equations, we select the diam-
D. Nagelhout et al.
/
eter of the cylinders D as the reference length and u(, as the reference velocity. Then the reference time is pD2/ 7, and the reference pressure is quo/D. With the above scaling, Eqs. (3) and (4) become
(5)
ug+u$
Re
[
I
205
Materials Science and Engineering A191 (1995) 203-208
ap =--+vv?u ay
where the Reynolds number is defined as Re = Du,,p/q and the variables are non-dimensional. The numerical solution to this boundary value problem was obtained using a finite element algorithm based on bilinear isoparametric elements and a Petrov-Galerkin formulation for convection [ 191. A penalty function is used to impose incompressibility. A time-dependent formulation is used to approach and approximate the steady-state. Diffusion and pressure terms are treated implicitly, while convective terms are explicit.
cylinder domain for the same fraction of liquid (gL= 0.9) yields a pressure drop over the second cylinder that is within 0.33% of that obtained using four cylinders in the domain. Therefore, three cylinders were employed in all subsequent calculations. For the lower Reynolds numbers, solving the system with a uniform velocity at the inlet was adequate for the flow to become periodic before it reached the second cylinder. However, for the higher Reynolds numbers. the flow was not periodic after passing over only one cylinder. We chose to use the velocity field between the second and third cylinders as the boundary condition at the inlet for the next computation. In other words. for the first computation the velocity was uniform (u =uJ at x= 0. The velocity at x =4L from the first computation was used as input at x= 0 for the next, and so on. We found that two iterations were adequate for Re = 40, while even after three iterations the results did not converge when g,_2 0.90 when Re = 95. Figs. 3(a) and 3(b) show the velocity vectors and the pressure contours for the case of Re = 5 and gL= 0.99. In Fig. 3(a) it can be seen that the flow field in the vicinity of and above the first cylinder differs from the flow field associated with the second and third cylinders. Periodicity can be observed, however, in the flow past the second and third cylinders. Fig. 3(b) also
3. Methods of calculation In order to find permeabilities for the cases that interested us, we first needed to compute pressure drops across cylinders using the algorithm and boundary conditions described previously. A typical mesh used in the calculations is shown in Fig. 2. The resulting permeabilities were verified by comparison with those obtained by Sangani and Acrivos [ 121 and Edwards et al. [ 181. However, Edwards et al. enforced periodicity in the flow, whereas in our method we imposed a uniform velocity ( ucl = 1) at the inlet (X = 0). Hence, it was necessary to determine how many cylinders had to be included in the computational domain to develop periodic flow. For the range of Reynolds numbers considered in this work, a three-cylinder domain suffices to develop a periodic flow over the second cylinder, This was validated with g,= 0.9 using a four-cylinder domain at a low Reynolds number (Re = 0.1). The average pressure obtained halfway between cylinders yields a 0.004% difference in the pressure drop over the third cylinder when compared with the pressure drop over the second cylinder, and this small difference is therefore negligible. The pressure drops over the first and fourth cylinders are not examined because they are affected by the inlet and outlet boundary conditions. These results suggested that a four-cylinder domain is unnecessarily large, and we determined that a three-
5
T i
%
0
10
5
20
15
x/D
Fig. 2. Computational domain used for the numerical of flow normal to circular cylinders in a square array.
5 -L~l~zL~
simulation
2L-LL-I
I-
T -,+.. 0
’ 0
5
_..--.-____ ______
_‘,/-; r 10
-_----___ ,.-‘--T--i-. 15
_;,-. 20
._______^ ____- i r 25
x/D
T i x/D
Fig. 3. Calculated results for Re = 5 and g,= 0.99: (a) velocity vectors, (b) pressure contours.
D. Nagelhout et al.
206
/
Materials Science and Engineering A191 (1995) 203-208
shows that, near and above the second and third cylinders, the pressure contours are essentially identical. The major goal of the computations is to determine the permeability as defined by Eq. (l), which relates the superficial velocity to the pressure gradient. In this case, the superficial velocity is precisely equal to the uniform velocity specified at the inlet. The pressure gradient is calculated from the difference in the average pressures at the plane midway between the second and third cylinders &, and between the first and second cylinders p,. The pressure gradient in Eq. ( 1 ), therefore, is approximated by
dF P2-P,
vp=z=2L
Having ascertained the pressure gradient, the permeability can be calculated with Eq. (2).
4. Results and discussion The calcualted pressure gradient for Stokes flow (Re = 0) as a function of the volume fraction of solid is shown in Fig. 4 and compared with the calculated results of Sangani and Acrivos [ 121 and the numerical results of Edwards et al. [ 181. The pressure gradients calcualted by the three sets of workers agree within 1.5% of each other for g, up to 0.5. At g,_=O.6 the agreement is also very good, with the two sets of numerical results approximately 6% greater than the results of Sangani and Acrivos [ 121.
In Fig. 5, we show pressure gradients calculated for dilute concentrations of solid for 0 I Re140. It is well known that flow past a single circular cylinder becomes unstable to small perturbations somewhere between Re = 30 and Re = 40, first affecting the wake at some distance downstream from the cylinder [20]. Furthermore, as the Reynolds number is increased to about Re = 60, a vortex sheds behind the cylinder. Accompanying the instability of the flow is a force normal to the cylinder axis [20], which indicates that the flow is not two-dimensional. Hence, our assumption of twodimensional flow is probably not strictly valid for Reynolds numbers greater than 30. The observations for a single cylinder are relevant, especially for very dilute mixtures (g, > 0.9) where the cylinders downstream are in the path of the unstable wake generated by the cylinders upstream. Therefore, the calculated results at Reynolds number greater than 40 are deemed to be incorrect because of the instability of the flow and because the numerical method lacks the necessary resolution to capture properly the unsteady fluid motion. Results at Re = 95 and g,Z0.9 did not converge to a steady state, and show a decrease in the pressure gradient when compared with the results for 01 Re140. Therefore, they show a permeability increase with an increase in Re, contrary to physical expectations. Strictly applied, Darcy’s law is valid for Stokes flow, and it is well known that inertial effects associated with finite values of a Reynolds number can cause a deviation in the proportionality between the velocity and the pressure gradient. Nevertheless, Edwards et al. [ 181 defined the permeability in Eq. (1) as the apparent
0
Volume
Fraction
Sohd
Fig. 4. Pressure gradient as a function of volume fraction of solid for Stokes flow (Re = 0).
I
L 0
10
20
Reynolds
30
40
50
Number
Fig. 5. Non-dimensional pressure gradient Reynolds number and volume fraction liquid.
as a function
of
/
D. Nagelhout et al.
1
Materials Science and Engineering A191 (I 995) 203-208
gL
=080
.
gL
=085
v
gL
=090
.
g,
=092
0
207
range 0.4 1gr10.8 are from Edwards et al. [ 181, and those in the range 0.8 I g,I 0.99 are calculated in this work. We see that the two works give consistent results. Although there is an effect of the Reynolds number, it is relatively small and certainly much smaller than the accuracy at which the permeability can be measured in dendritic structures [lo].
5. Conclusions
1
1
5
”
15
10
20
25
Reynolds
Fig. 6. Normalized permeability liquid and Reynolds number.
30
35
40
45
50
Number
as a function of volume fraction
7
t
From the present study the following conclusions can be drawn. ( 1) The non-dimensional permeability for both Stokes flow and flow with a finite Reynolds number was calculated with a finite element computer simulation. The calculated results agree very well with the analytical results of Sangani and Acrivos [ 121 and with the numerical computations of Edwards et al. [ 181. For finite Reynolds numbers, we have extended the results to volume fractins of liquid as high as 0.99, in order to achieve relevancy to the solidification of alloys and to the infiltration of fiber composites. (2) A dependence of the non-dimensional permeability on the Reynolds number has been observed, but this is not a strong dependence. (3) The present study confirms the fact that in flows with a low Reynolds number, the assumption of steady flow at very high volume fractions of liquid in a square array of cylinders is valid.
Acknowledgments
The authors are grateful for the sponsorship of the National Science Foundation (NSF grant DMR9111106). -3
-0
/ 4
-0
3
-0
2
-0
1
00
References
log g, Fig. 7. The apparent fraction of fluid g,
permeability
as a function
of the volume
permeability for the purpose of analyzing the behavior of the flow as a function of the Reynolds number. Fig. 6 gives our calculated results for 0.8ig,_10.99 in terms of the normalized permeability, which is defined as K/Z& where K is the apparent permeability and K, is the permeability for Stokes flow (Re = 0). The results for the scope of 0.2
[II A. Mortensen,
L.J. Masur, J.A. Cornie and M.C. Flemings, Infiltration of fibrous preforms by a pure metal: part 1, theory, Metall. Trans. A, 20 (1989) 25352547. L-21J.A. Sekhar and R. Trivedi, Development of solidification microstructures in the presence of fibers or channels of finite width, Mater. Sci. Eng., All4 (1989) 133-146. and M.F. [31 S.M. Copley, A.F. Giamei, SM. Johnson Hornbecker, The origin of freckles in unidirectionally solidified castings, Metall. Trans., 1 (1970) 2193-2204. [41 R. Mehrabian, M.C. Keane and M.C. Flemings, Interdendritic fluid flow and macrosegregation: influence of gravity, Metall. Trans., 1 (1970) 1209-1220. PI J.R. Sarazin and A. Hellawell, Channel formation in Pb-Sn, Pb-Sb and Pb-Sn-Sb alloy ingots and comparisons with the system NH,Cl-H,O, Metall. Trans. A, 19 (1988) 1861-1871.
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D. Nagelhout et al.
[61 W.D. Bennon and F.P. Incropera,
/
Materials Science and Engineering A 191 (1995) 203-208
The evolution of macrosegregation in statically cast binary ingots, Metall. Trans. B, 18(1987)611-616. 171 C. Beckermann and R. Viskanta, Double-diffusive convection during dendritic solidification of a binary mixture, Phys. Chem. Hydrodyn., IO (1988) 195-213. [81 SD. Felicelli, J.C. Heinrich and D.R. Poirier, Simulation of freckles during vertical solidification of binary alloys, Metall. Trans. B, 22 (1991) 847-859. [91 G.H. Geiger and D.R. Poirier, Trunsport Phenomena in Metallurgy, Addison-Wesley, Reading, MA, 1973, p. 9 1. [lOI D.R. Poirier, Permeability for flow of interdendritic liquid in columnar-dendritic alloys. Merall. Trans. B, 19 (1989) 245-255. [Ill S. Ganesan, C.L. Chan and D.R. Poirier, Permeability for flow parallel to primary dendrite arms, Mater. Sci. Eng., A151 (1992) 97-105. [I21 AS. Sangani and A. Acrivos, Slow flow past periodic arrays of cylinders with application to heat transfer, ht. J. Multiphase Flow, 8 (1982) 193-206. [I31 E.M. Sparrow and A.L. Loefler, Longitudinal laminar flow between cylinders arranged in regular array, AIChE .I., 5 (1959) 325-330. [I41 R.E. Larson and J.J.L. Higdon, Microscopic flow near the surface of two-dimensional porous media, part 1, Axial flow, J. Fluid Mech., 166 (1986) 449-472. [I51 B.E. Launder and T.H. Massey, The numerical prediction of viscous flow and heat transfer in tube banks, J. Heat Transfer, 100 (1978) 565-571. [161 A. Thorn and C.J. Apelt, Field Computations in Engineering and Physics, Van Nostrand, London, 196 1, pp. 123-138. R.G. Carbonell, S. Whitaker and L.R. [I71 A. Eidseth, Hermann, Dispersion in pulsed systems--III comparison between theory and experiments for packed beds, Chem. Eng. Sci., 38 (1983) 1803-1816. 1181 D.A. Edwards, M. Shapiro, P. Bar-yoseph and M. Shapira, The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders, Phys. Fluids A, 2 (1992) 45-55. [I91 J.C. Heinrich and C.C. Yu, Finite element simulation of buoyancy-driven flows with emphasis on natural convection in a horizontal circular cylinder, Comput. Methods Appl. Mech. Eng., 69 (1988) l-27.
[20] G.K. Batchelor, An introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967, pp. 260-261.
Appendix A: Derivation of Eq. (2)
We start with Darcy’s law with a one-directional superficial velocity:
7 L
(Al)
Let u(, be the reference velocity, the diameter of the cylinders D be the reference length, and qu,,/D the reference pressure, then Eq. (Al) can be written
U’=$VP’
L42)
where the variables with primes are non-dimensional and those without primes are dimensional. When dealing with microstructures, it is convenient to select the specific surface of the solid as the characteristic length for the permeability. Hence, the nondimensional permeability is KSv2. When the solid comprises circular cylinders of uniform diameter, then S, = 4/O and Eq. (A2) becomes KS,’ = ~1617’ VP’
(A3)
The non-dimensional velocity U’ and the non-dimensional pressure gradient, Vp’ = Ap’/L’, are obtained as the output from the computer simulation so that KS,’ can be calculated.
Materials Science and Engineering
Al 91 (1995) 203-208
Permeability for flow normal to a sparse array of fibres D. Nagelhouta, M.S. Bhatb, J.C. Heinricha, D.R. Poirierb “Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 8.5721, USA bDepartment of Materials Science and Engineering, University ofArizona, Tucson, AZ 85721, USA Received 7 March 1994; in revised form 27 July 1994
Abstract Solutions for flow normal to an array of fibers (visualized as circular cylinders) arranged in a square array were determined using a finite element formulation. The calculated results were used to obtain the permeability for flow through the system with as little as 0.01 volume fraction solid and for a Reynolds number up to 40. These results can be used to estimate permeabilities for infiltrating fiber composites or for modeling the solidification of dendritic alloys with transport phenomena in porous media. Keywords: Fibres; Permeability
1. Introduction
p-Kvp I
Ever since fiber composites were produced by the liquid metallurgy route, their solidification behavior and microstructural development have been the subject of intense research. Mortensen et al. [l] used Darcy’s law to analyze the infiltration of metal through fibers for the production of metal matrix composites. Recently Sekhar and Trivedi [2] reported work on the interaction between fibers, narrow channels, and solidification fronts in a transparent analog system. Their study provides clear evidence for the strong effect of fiber arrangements on microsegregation patterns. It is also known that the convection of the interdendritic liquid through the dendritic mushy zone in solidifying alloys is responsible for most forms of macrosegregation in castings and ingots [3-51. The nature and extent of the segregation can be modeled as transport phenomena in porous media [6-81; therefore, to model the convection and its attendant macrosegregation in solidifying alloys, either the permeability as found in Darcy’s law or a drag coefficient derived from the permeability is required. According to Darcy’s law, the seepage velocity (also called the superficial velocity) U and the applied pressure gradient Vp are related by the expression [9], 0921-5093/95/$9.50 0 1995 - Elsevier Science S.A. All rights reserved SSD/O921-X)93(94)09641-4
I
11
where 11is the viscosity of interdendritic liquid and K is the permeability of the mushy zone. The permeability itself relates directly to the microstructural length scale of the solid comprising the dendritic network or fiber array and to the volume fraction of liquid. Utilizing these metrics, Darcy’s law can be rearranged to obtain the dimensionless permeability KS,’ 16U KS,’ = ~ VP
(2)
where S, is the specific surface area of the solid, U has been converted to the non-dimensional seepage velocity and Vp to the non-dimensional pressure gradient. Eq. (2) is derived in Appendix A. In most porous media, the fluid phase convects in a network of a uniform volume fraction of solid. However, in a solidifying alloy the interdendritic liquid moves in a solid network in which the volume fraction of liquid g, is not uniform. In the interior of the mushy zone (e.g. where 0 Ig,50.7), the momentum equation can be satisfactorily represented by Eq. ( 1). In the context of dendritic solidification, however, Eq. (1)
204
LI. Nagelhout et al.
/
Materials Science and Engineering A191 (199.5)203-208
gives erroneous results in the part of the mushy zone that is adjacent to the all-liquid zone, where g, exceeds 0.7, and it is necessary to use a comprehensive momentum equation with inertial terms and diffusion terms, along with the Darcy term. Also, the complete momentum equation should provide for a smooth transition of the momentum transport, from the mushy zone to the all liquid zone [S]. Specifically, as g,- 1, K -+ ~0 and the momentum equation reduces to the Navier-Stokes equation for a single-phase liquid. In order to characterize the permeabilities in dendritic solidification processes, it is necessary to consider flow in three categories: (1) flow through a columnar dendritic network with flow parallel to the primary dendrite arms; (2) flow through a columnar dendritic network with flow normal to the primary dendrite arms; (3) flow through a network of equiaxial grains. This paper addresses the second flow situation. There are no empirical data for permeability for flow through columnar dendritic networks when g,_> 0.66, and permeabilities cannot be measured at higher values of g, [lO,ll]. Therefore, we rely on direct numerical simulations of flow through dendritic microstructures and analytical results of flow through “model” microstructures. Sangani and Acrivos [ 121, Sparrow and Loefler [ 131, and Larson and Higdon [14] calculated the drag force for Stokes flow through two-dimensional periodic arrays of circular cylinders. Also, Launder and Massey [ 151, Thorn and Apelt [ 161 and Eidseth et al. [ 171 used numerical methods to calculate drag forces and permeabilities of flows with finite Reynolds numbers, but in these works the porosities were limited to a maximum of 0.5. More recently Edwards et al. [18] analyzed flows for Reynolds number up to 180, while extending the porosity to 0.8. For application to solidification scenarios, however, the volume fraction of liquid goes all the way to unity, and infiltrated fiber composites also contain less than 0.2 volume fraction of fibers. Hence, the present paper aims at calculating the permeability at a very high volume fraction of fluid, for which no empirical data are available. This is accomplished using a numerical method, based on finite elements, to calculate the two-dimensional flow normal to circular cylinders arranged in square lattices. By altering the ratio of diameter to the center-to-center distance between cylinders, the porosity can be made as high as 0.99.
> >
>
U Fig. 1. Flow through a square array of circular cylinders.
au
au
ap
ug+uaL'=-~+q
(3)
and
a% a% ,,a”+,atl_ap+ vg+ayz ax ay ay i i subject to the incompressibility
For the two-dimensional flow through the array shown in Fig. 1, the components of the Navier-Stokes equation are
constraint
au+!!%0
ax ay
where u and u are the x- and ycomponents of the velocity respectively, p is pressure, and q is the viscosity. A computational domain with three semicylinders is shown as Fig. 2. The cylinders have a radius R, and the center-to-center distance between them is 2L. Flow is symmetrical at the planes y= 0 and y= L. The boundary conditions for velocity are as follows: v =0
at
y= 0
v =0
at
y=L
and u = ug (uniform)
at
Along the solid-liquid sure we simply specify 2. Formulation
(4)
x=0 interfaces,
u =U = 0. For pres-
p( 0, L ) = p. (constant)
The numerical solutions are obtained in non-dimensional form using a finite element algorithm. In order to non-dimensionalize the equations, we select the diam-
D. Nagelhout et al.
/
eter of the cylinders D as the reference length and u(, as the reference velocity. Then the reference time is pD2/ 7, and the reference pressure is quo/D. With the above scaling, Eqs. (3) and (4) become
(5)
ug+u$
Re
[
I
205
Materials Science and Engineering A191 (1995) 203-208
ap =--+vv?u ay
where the Reynolds number is defined as Re = Du,,p/q and the variables are non-dimensional. The numerical solution to this boundary value problem was obtained using a finite element algorithm based on bilinear isoparametric elements and a Petrov-Galerkin formulation for convection [ 191. A penalty function is used to impose incompressibility. A time-dependent formulation is used to approach and approximate the steady-state. Diffusion and pressure terms are treated implicitly, while convective terms are explicit.
cylinder domain for the same fraction of liquid (gL= 0.9) yields a pressure drop over the second cylinder that is within 0.33% of that obtained using four cylinders in the domain. Therefore, three cylinders were employed in all subsequent calculations. For the lower Reynolds numbers, solving the system with a uniform velocity at the inlet was adequate for the flow to become periodic before it reached the second cylinder. However, for the higher Reynolds numbers. the flow was not periodic after passing over only one cylinder. We chose to use the velocity field between the second and third cylinders as the boundary condition at the inlet for the next computation. In other words. for the first computation the velocity was uniform (u =uJ at x= 0. The velocity at x =4L from the first computation was used as input at x= 0 for the next, and so on. We found that two iterations were adequate for Re = 40, while even after three iterations the results did not converge when g,_2 0.90 when Re = 95. Figs. 3(a) and 3(b) show the velocity vectors and the pressure contours for the case of Re = 5 and gL= 0.99. In Fig. 3(a) it can be seen that the flow field in the vicinity of and above the first cylinder differs from the flow field associated with the second and third cylinders. Periodicity can be observed, however, in the flow past the second and third cylinders. Fig. 3(b) also
3. Methods of calculation In order to find permeabilities for the cases that interested us, we first needed to compute pressure drops across cylinders using the algorithm and boundary conditions described previously. A typical mesh used in the calculations is shown in Fig. 2. The resulting permeabilities were verified by comparison with those obtained by Sangani and Acrivos [ 121 and Edwards et al. [ 181. However, Edwards et al. enforced periodicity in the flow, whereas in our method we imposed a uniform velocity ( ucl = 1) at the inlet (X = 0). Hence, it was necessary to determine how many cylinders had to be included in the computational domain to develop periodic flow. For the range of Reynolds numbers considered in this work, a three-cylinder domain suffices to develop a periodic flow over the second cylinder, This was validated with g,= 0.9 using a four-cylinder domain at a low Reynolds number (Re = 0.1). The average pressure obtained halfway between cylinders yields a 0.004% difference in the pressure drop over the third cylinder when compared with the pressure drop over the second cylinder, and this small difference is therefore negligible. The pressure drops over the first and fourth cylinders are not examined because they are affected by the inlet and outlet boundary conditions. These results suggested that a four-cylinder domain is unnecessarily large, and we determined that a three-
5
T i
%
0
10
5
20
15
x/D
Fig. 2. Computational domain used for the numerical of flow normal to circular cylinders in a square array.
5 -L~l~zL~
simulation
2L-LL-I
I-
T -,+.. 0
’ 0
5
_..--.-____ ______
_‘,/-; r 10
-_----___ ,.-‘--T--i-. 15
_;,-. 20
._______^ ____- i r 25
x/D
T i x/D
Fig. 3. Calculated results for Re = 5 and g,= 0.99: (a) velocity vectors, (b) pressure contours.
D. Nagelhout et al.
206
/
Materials Science and Engineering A191 (1995) 203-208
shows that, near and above the second and third cylinders, the pressure contours are essentially identical. The major goal of the computations is to determine the permeability as defined by Eq. (l), which relates the superficial velocity to the pressure gradient. In this case, the superficial velocity is precisely equal to the uniform velocity specified at the inlet. The pressure gradient is calculated from the difference in the average pressures at the plane midway between the second and third cylinders &, and between the first and second cylinders p,. The pressure gradient in Eq. ( 1 ), therefore, is approximated by
dF P2-P,
vp=z=2L
Having ascertained the pressure gradient, the permeability can be calculated with Eq. (2).
4. Results and discussion The calcualted pressure gradient for Stokes flow (Re = 0) as a function of the volume fraction of solid is shown in Fig. 4 and compared with the calculated results of Sangani and Acrivos [ 121 and the numerical results of Edwards et al. [ 181. The pressure gradients calcualted by the three sets of workers agree within 1.5% of each other for g, up to 0.5. At g,_=O.6 the agreement is also very good, with the two sets of numerical results approximately 6% greater than the results of Sangani and Acrivos [ 121.
In Fig. 5, we show pressure gradients calculated for dilute concentrations of solid for 0 I Re140. It is well known that flow past a single circular cylinder becomes unstable to small perturbations somewhere between Re = 30 and Re = 40, first affecting the wake at some distance downstream from the cylinder [20]. Furthermore, as the Reynolds number is increased to about Re = 60, a vortex sheds behind the cylinder. Accompanying the instability of the flow is a force normal to the cylinder axis [20], which indicates that the flow is not two-dimensional. Hence, our assumption of twodimensional flow is probably not strictly valid for Reynolds numbers greater than 30. The observations for a single cylinder are relevant, especially for very dilute mixtures (g, > 0.9) where the cylinders downstream are in the path of the unstable wake generated by the cylinders upstream. Therefore, the calculated results at Reynolds number greater than 40 are deemed to be incorrect because of the instability of the flow and because the numerical method lacks the necessary resolution to capture properly the unsteady fluid motion. Results at Re = 95 and g,Z0.9 did not converge to a steady state, and show a decrease in the pressure gradient when compared with the results for 01 Re140. Therefore, they show a permeability increase with an increase in Re, contrary to physical expectations. Strictly applied, Darcy’s law is valid for Stokes flow, and it is well known that inertial effects associated with finite values of a Reynolds number can cause a deviation in the proportionality between the velocity and the pressure gradient. Nevertheless, Edwards et al. [ 181 defined the permeability in Eq. (1) as the apparent
0
Volume
Fraction
Sohd
Fig. 4. Pressure gradient as a function of volume fraction of solid for Stokes flow (Re = 0).
I
L 0
10
20
Reynolds
30
40
50
Number
Fig. 5. Non-dimensional pressure gradient Reynolds number and volume fraction liquid.
as a function
of
/
D. Nagelhout et al.
1
Materials Science and Engineering A191 (I 995) 203-208
gL
=080
.
gL
=085
v
gL
=090
.
g,
=092
0
207
range 0.4 1gr10.8 are from Edwards et al. [ 181, and those in the range 0.8 I g,I 0.99 are calculated in this work. We see that the two works give consistent results. Although there is an effect of the Reynolds number, it is relatively small and certainly much smaller than the accuracy at which the permeability can be measured in dendritic structures [lo].
5. Conclusions
1
1
5
”
15
10
20
25
Reynolds
Fig. 6. Normalized permeability liquid and Reynolds number.
30
35
40
45
50
Number
as a function of volume fraction
7
t
From the present study the following conclusions can be drawn. ( 1) The non-dimensional permeability for both Stokes flow and flow with a finite Reynolds number was calculated with a finite element computer simulation. The calculated results agree very well with the analytical results of Sangani and Acrivos [ 121 and with the numerical computations of Edwards et al. [ 181. For finite Reynolds numbers, we have extended the results to volume fractins of liquid as high as 0.99, in order to achieve relevancy to the solidification of alloys and to the infiltration of fiber composites. (2) A dependence of the non-dimensional permeability on the Reynolds number has been observed, but this is not a strong dependence. (3) The present study confirms the fact that in flows with a low Reynolds number, the assumption of steady flow at very high volume fractions of liquid in a square array of cylinders is valid.
Acknowledgments
The authors are grateful for the sponsorship of the National Science Foundation (NSF grant DMR9111106). -3
-0
/ 4
-0
3
-0
2
-0
1
00
References
log g, Fig. 7. The apparent fraction of fluid g,
permeability
as a function
of the volume
permeability for the purpose of analyzing the behavior of the flow as a function of the Reynolds number. Fig. 6 gives our calculated results for 0.8ig,_10.99 in terms of the normalized permeability, which is defined as K/Z& where K is the apparent permeability and K, is the permeability for Stokes flow (Re = 0). The results for the scope of 0.2
[II A. Mortensen,
L.J. Masur, J.A. Cornie and M.C. Flemings, Infiltration of fibrous preforms by a pure metal: part 1, theory, Metall. Trans. A, 20 (1989) 25352547. L-21J.A. Sekhar and R. Trivedi, Development of solidification microstructures in the presence of fibers or channels of finite width, Mater. Sci. Eng., All4 (1989) 133-146. and M.F. [31 S.M. Copley, A.F. Giamei, SM. Johnson Hornbecker, The origin of freckles in unidirectionally solidified castings, Metall. Trans., 1 (1970) 2193-2204. [41 R. Mehrabian, M.C. Keane and M.C. Flemings, Interdendritic fluid flow and macrosegregation: influence of gravity, Metall. Trans., 1 (1970) 1209-1220. PI J.R. Sarazin and A. Hellawell, Channel formation in Pb-Sn, Pb-Sb and Pb-Sn-Sb alloy ingots and comparisons with the system NH,Cl-H,O, Metall. Trans. A, 19 (1988) 1861-1871.
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The evolution of macrosegregation in statically cast binary ingots, Metall. Trans. B, 18(1987)611-616. 171 C. Beckermann and R. Viskanta, Double-diffusive convection during dendritic solidification of a binary mixture, Phys. Chem. Hydrodyn., IO (1988) 195-213. [81 SD. Felicelli, J.C. Heinrich and D.R. Poirier, Simulation of freckles during vertical solidification of binary alloys, Metall. Trans. B, 22 (1991) 847-859. [91 G.H. Geiger and D.R. Poirier, Trunsport Phenomena in Metallurgy, Addison-Wesley, Reading, MA, 1973, p. 9 1. [lOI D.R. Poirier, Permeability for flow of interdendritic liquid in columnar-dendritic alloys. Merall. Trans. B, 19 (1989) 245-255. [Ill S. Ganesan, C.L. Chan and D.R. Poirier, Permeability for flow parallel to primary dendrite arms, Mater. Sci. Eng., A151 (1992) 97-105. [I21 AS. Sangani and A. Acrivos, Slow flow past periodic arrays of cylinders with application to heat transfer, ht. J. Multiphase Flow, 8 (1982) 193-206. [I31 E.M. Sparrow and A.L. Loefler, Longitudinal laminar flow between cylinders arranged in regular array, AIChE .I., 5 (1959) 325-330. [I41 R.E. Larson and J.J.L. Higdon, Microscopic flow near the surface of two-dimensional porous media, part 1, Axial flow, J. Fluid Mech., 166 (1986) 449-472. [I51 B.E. Launder and T.H. Massey, The numerical prediction of viscous flow and heat transfer in tube banks, J. Heat Transfer, 100 (1978) 565-571. [161 A. Thorn and C.J. Apelt, Field Computations in Engineering and Physics, Van Nostrand, London, 196 1, pp. 123-138. R.G. Carbonell, S. Whitaker and L.R. [I71 A. Eidseth, Hermann, Dispersion in pulsed systems--III comparison between theory and experiments for packed beds, Chem. Eng. Sci., 38 (1983) 1803-1816. 1181 D.A. Edwards, M. Shapiro, P. Bar-yoseph and M. Shapira, The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders, Phys. Fluids A, 2 (1992) 45-55. [I91 J.C. Heinrich and C.C. Yu, Finite element simulation of buoyancy-driven flows with emphasis on natural convection in a horizontal circular cylinder, Comput. Methods Appl. Mech. Eng., 69 (1988) l-27.
[20] G.K. Batchelor, An introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967, pp. 260-261.
Appendix A: Derivation of Eq. (2)
We start with Darcy’s law with a one-directional superficial velocity:
7 L
(Al)
Let u(, be the reference velocity, the diameter of the cylinders D be the reference length, and qu,,/D the reference pressure, then Eq. (Al) can be written
U’=$VP’
L42)
where the variables with primes are non-dimensional and those without primes are dimensional. When dealing with microstructures, it is convenient to select the specific surface of the solid as the characteristic length for the permeability. Hence, the nondimensional permeability is KSv2. When the solid comprises circular cylinders of uniform diameter, then S, = 4/O and Eq. (A2) becomes KS,’ = ~1617’ VP’
(A3)
The non-dimensional velocity U’ and the non-dimensional pressure gradient, Vp’ = Ap’/L’, are obtained as the output from the computer simulation so that KS,’ can be calculated.