Axisymmetric flow near a rotating permeable bed

FLUIDDYNAMICS RESEARCH

ELSEVIER

Fluid Dynamics Research 16 (1995) 147-160

Axisymmetric flow near a rotating permeable bed K.N. Mehta, K. N a r a s i m h a Rao Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India Received 8 October 1992; revised 17 May 1993

Steady-state boundary layer flow generated on a rotating fluid-saturated porous bed when a column of axisymmetric frictionless potential flow arriving from infinity impinges on it is studied numerically by employing an appropriate boundary condition to account for the presence of tarIgential slip in the radial and transverse directions on the permeable interface which separates the boundary layer from the Darcy flow occurring in the porous bed. Two important physical parameters which influence the dynamic coupling between these two flows are rotation and permeability of the porous bed. An interesting result of this study is that to each value of rotation parameter s, there corresponds a permeability value K o for which the velocity component normal to the bed and the radial shear stress vanish at every point on the bed; that is, under certain conditions the radial shear stress can be completely annihilated and the porous bed made to behave as if it were an impermeable plane.

Notations f,g f"(0)

g'(0) prime (') F k K P P r, O, Z

R,Z s U, V, W

U,V,W Zo

dimensionless velocity functions dimensionless measure of radial shear stress dimensionless measure of transverse shear stress denotes derivative with respect to Z dimensionless pressure function permeability dimensionless permeability parameter fluid pressure dimensionless pressure cyclindrical polar coordinates dimensionless radial and axial coordinates dimensionless rotation control parameter radial, transverse, and axial velocity components non-dimensional radial, transverse, and axial velocity components location of point of flow reversal

0169-5983/95 / $04.50 © 1995 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved SSDI 0 1 6 9 - 5 9 8 3 ( 9 5 ) 0 0 0 0 7 - 0

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147 160

148

Greek symbols #, v f2~

dimensionless constant introduced in Eq. (15) coefficients of dynamic and kinematic viscosity characteristic angular speed

Subscripts oo, d

refer to quantities in potential flow and Darcy flow

I Introduction

Flows through porous media and past stationary or rotating porous surfaces are prevalent in nature and of principal interest in scientific and engineering applications. Knowledge of the details of flows past rotating porous boundaries is of practical significance with regard to problems of gaseous diffusion boundary cooling, centrifuges, and filtration process (Zandbergen and Dijkstra, 1987; Elkouh, 1970). Also it has important applications in the lubrication of externally pressurized thrust bearings (Wang, 1979). Steady flow of an incompressible viscous liquid due to an infinite rotating disk was first discussed by Von Karman (1921), who obtained an approximate solution to the problem using the integral method, while Cochran (1934) corrected his solution and found more accurate results by numerical treatment of the equations. B6dewadt (1940) solved numerically the related problem of the flow produced over an infinite stationary plane in a fluid which is rotating with uniform angular velocity at an infinite distance from the plane. Both these flows are particular cases of the general family of rotationally symmetric flows described qualitatively by Batchelor (1951). The problem has also been studied by Stewartson (1953). Stuart (1954) integrated Karman's equations and gave a quantitative description of the effect of the suction at the porous boundary. Based on the basic work of Von Karman (1921), which examined self-similar solution of the complete Navier-Stokes equations for flow above a rotating disk, a complete discussion devoted to the study of this and other related flows can be found in the comprehensive reviews by Moore (1956), Parter (1982) and Zandbergen and Dijkstra (1987). Flow near a porous rotating disk has been studied by many researchers (see references quoted in Zandbergen and Dijkstra, 1987) by modelling it as a coupled boundary value problem with uniform suction or injection at the interface between fluid and porous medium. But a naturally permeable body differs from one for which mass removal is prescribed in that flow conditions at the porous surface cannot be determined a priori but depend on the nature of the field flow within the porous matrix as well as the flow exterior to it (Joseph, 1965). The present study is motivated by the work of Joseph (1965), who studied the problem of flow induced by rotation of a naturally permeable disk. Joseph (1965) identified the suction parameter introduced by Stuart (1954) with a permeability parameter which depends on the Darcy coefficient and other given data. The present work examines dynamic coupling between Darcy flow and boundary layer flow produced over an infinite rotating fluid-saturated porous bed as a result of an axisymmetric potential flow arriving from infinity and impinging on the porous bed. In our problem, the role of porosity is considered in a more realistic manner by taking into account not only a non-zero

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160

149

component of velocity normal to the bed but also the presence of tangential slip on the bed along the radial and transverse directions. Analysis of flows past a fluid-saturated porous surface requires the selection of an appropriate boundary condition for a realistic description of tangential velocity at the permeable boundary. Beavers and Joseph (1967) proposed a slip boundary condition to deal with such flow situations. Subsequent experiments (Beavers et al., 1970, 1974; Taylor, 1971; Sparrow, 1973) in channels with porous boundaries gave results which closely agreed with the theoretical predictions obtained using this boundary condition. Later, Saffman (1971) provided a sound theoretical justification for the validity of the slip boundary condition by an analytical derivation based on the idea of limiting case of step function distribution of permeability of the porous matrix. In recent papers Mehta (1984, 1991, 1994) and Mehta and Narasimha Rao (1992, 1993, 1994) used the slip boundary condition to study transient flows in cavities of rectangular and circular cross-sections embedded in fluid-saturated non-erodible porous media of uniform porosity. This paper is a sequel to the earlier work of Mehta and Narasimha Rao (1993, 1994), which dealt with the dynamic coupling of boundary layer on a stationary saturated porous bed with Darcy low when the flow impinging on the bed is either a plane two-dimensional or an axisymmetric potential flow. The main aim of this paper is to examine the influence of rotation and permeability on some important features of boundary layer flow on a porous bed. The problem is formulated under the assumptions that a similarity solution exists and that the bed is made up of a cylindrical porous matrix of infinite radius. Consequently, any edge effects arising from the finite radial extent of the rotating porous bed may be neglected.

2. Statement of the problem and mathematical formulation Let a column of axisymmetric frictionless potential flow arriving from infinity (z = ~) come in contact with an infinitely extending fluid-saturated porous bed rotating with uniform angular velocity f2 = st2+ about the z-axis chosen normal to the bed (Fig. 1). Here f2oo denotes the characteristic angular velocity used to describe the potential flow at infinity and s is a scalar which controls the angular velocity of the bed. Cylindrical polar coordinates (r, 0, z) are used with the porous bed occupying the half-space z ~< 0 while the region z > 0 is occupied by the fluid impinging on the bed. The axisymmetric potential flow at z = oo is described by u+, v~, w+ = rl2~,O, -- 2zf2~.

(1)

It can be readily verified that the flow field (1) is rotation-free. Then pressure p+ of the potential flow is given by Bernoulli's equation po - p +

2

(2)

= ½p(u£ + w+),

in which p denotes fluid density and Po, the pressure at the point r = 0, z = 0. It is convenient to introduce the following dimensionless variables: r

z

u

v

w

p

(3)

150

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160

fiat

.

.

.

.

.

°

-

. , °

Fig. 1. Flow visualization.

Then the potential flow field in dimensionless form is described by U®, Voo, W~o = R, 0, - 2 Z

(4)

Po - Po~ = R 2 + 4 Z 2.

(5)

The flow field in the surface boundary layer on the rotating bed is assumed in the form U, V, W = R f ' (Z), Ro(Z), - 2 f ( Z ) ,

(6)

P0 - P = R2 + 4F(Z).

(7)

The relation (6) for velocity field satisfies the equation of continuity. On using Eqs. (6) and (7), the Navier-Stokes equations (see Schlichting, 1979: 66) lead to the following nonlinear system of equations for determining the unknown functions f, g, and F: f,,, = f , 2 _ 2 f f "

- 92 - 1,

9" = 2 ( f ' g - f g'), F' = f " + 2 f f ' .

(8) (9) (10)

The definition of Po implies F(0) = 0. Hence the first integral of Eq. (10) determines the pressure function as F (Z) -- f ' (Z) + f 2 ( Z ) - f ' ( 0 ) - f 2 ( 0 ) .

It should be noted that F depends implicity on 9.

(11)

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151

As soon as the potential flow comes in contact with the bed, the viscosity of the fluid becomes operative. Due to rotation and tangential slip at the permeable interface, the fluid in the boundary layer is thrown radially outwards while the potential flow slides over it. Unlike in the case of an impermeable bed where no slip condition holds, suitable boundary conditions have to be employed to account for radial and transverse tangential slip at the permeable interface Z = 0 for the problem under study. On account of transfer of momentum by seepage from the boundary layer into the porous bed, Darcy flow described by k 0p k 0p Or, -- - - # 0r' # 0z

Ud, Vd, Wd --

(12)

is generated just below the interface Z = 0. Here # denotes the coefficient of dynamic viscosity of fluid and k, the permeability of the porous bed. Since Darcy flow is solely determined by pressure gradient, its dynamic interaction with boundary layer flow is studied by invoking the continuity of radial and axial pressure gradients at every point on the plane Z = 0, that is,

OP (R, 0 --) 0R

=

OP

(R, 0) =

0P (R, 0 - ) = OP 0Z ~-~ (R, 0) -

OP

(R, 0 + ),

(13)

OP 0Z (R, 0 + ).

(14)

Using Eqs. (7), (12), (14), the Darcy flow can be expressed in dimensionless form as

Ud, I'd, Wd = K2R, sR, 2K2F'(0),

(15)

in which K denotes the dimensionless permeability number defined by K = k ~ / v . Noting that W (R, 0 - ) = Wd, the continuity of axial velocity across Z = 0, namely W (R, 0 - )

= W (R, O) = W (R, 0 +)

(16)

yields f(0) = - K 2 F ' ( 0 ) .

(17)

Following Beavers and Joseph (1967), the tangential slip along the radial and transverse directions on the porous bed is accounted for by the following boundary conditions for u and v: Ou 0z

~ X/~(V--Vd)'

0v ~=X//-~(V--Vo)

at

z=0,

which, in dimensionless form, become f'(0) = K 2 + K f " ( 0 ) ,

(18)

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152

K g(0) = s + -- 9'(0).

(19)

Here a is a dimensionless constant depending on the structure of the porous medium. We took = 1 while finding the numerical solution. Using Eq. (10) written at Z = 0 and Eq. (18), the boundary condition (17) takes the form f(O) =

KZf"(0) 1 + 2 K 4 + (2K3/a)f"(O)"

(20)

Since U ~ Uo~, V ---, 0, as Z ~ ~ functions f, g must also satisfy the asymptotic b o u n d a r y conditions f'--+l,

as

Z--+oo,

(21)

g--,O,

as

Z~

(22)

~.

Also, since W ~ W®, P ~ P~ and Z ~ ~ , f and F should behave like Z and Z z for large values of Z. This asymptotic property was used to check the accuracy of the numerical solution of the nonlinear b o u n d a r y value problem described by Eqs. (8) and (9) and the boundary conditions (18)-(22). This problem involves two physical parameters, s and K. The main aim of this paper is to examine the response of the dynamic interaction of b o u n d a r y layer flow with Darcy flow to variations in s and K.

3. Asymptotic behavior of solution for large Z Since f ' (Z) --+ 1 and g(Z) --+ 0 as Z ~ ~ we write, for large Z, f ( Z ) = Z + 2(Z),

g(Z) = #(Z),

where the functions 2(Z),/~(Z) are small for large Z and satisfy the conditions Z(oo) = 0, p(oo) -- 0. Substituting for f and g in Eqs. (8) and (9), and neglecting terms involving squares and products of 2, # and their derivatives, we find that 2' and/~ satisfies the linear h o m o g e n e o u s equation y" + 2Zy' -- 2y = 0,

y(oo) = 0.

(23)

It is obvious that one solution of the equation is y = Z, and this leads to the general solution y = CIZ + C2Z f;

exp ( ~ t2

t2)

dr,

where C1 and C2 are constants. All solutions of Eq. (23) which are b o u n d e d and vanish at Z = oo are given by the Ca solution, namely, y = CEZ f z e x p (t~- - t 2 ) dt. Hence, for large Z, the asymptotic expressions for 2'(Z) and #(Z) have the form (24).

(24)

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153

4. Numerical procedure Nonlinear occurrence of K in the boundary condition suggests the usefulness of looking for a perturbation series solution in powers of K, keeping s fixed, by writing f=fo

+ K f l + K 2 f 2 + ...,

g = go + K g l + K2g2 + ""

This leads to a nonlinear problem for fo, go and linear problems for fl, gl and f2, g2 involving predetermined variable coefficients. The numerical solution of these three problems will yield a series solution for f and g up to the terms involving K 2. This approach will involve much more computational effort than required in the direct numerical treatment of the original problem for f and g. In view of this observations, it was decided to solve the original problem for f and g numerically for a practical range of values of s and K by following the procedure outlined below. When the solution was computed, the range of values of s and K could be extended without encountering computational instability. The boundary value problem described by Eqs. (8) and (9) and boundary conditions (18)-(22) was descretized, resulting in the tridiagonal system AnUn-

1 "q- B n U n + C n U n + 1 =

(25)

D..

System (25) was obtained by approximating the nonlinear terms f ' g and f,2 by ;2 = ~J?l 9 ( ' Jn (,

.2

- - J (l l, . 2

f , g . = f,~. g. + f , g .

_ f,~. g . ,

where superscript • refers to the approximate value of the corresponding quantity. In Eq. (25) U. stands for ( f ' , g.) T while A., B., C., and D. are coefficient matrices given by

An[lJh2jh 0] Cn =

0

1/h 2 - f . / h

1/h2 + f . / h 0

0 1/h 2 + L / h

'

B. =

[2Jh22* g.

'

D. =

-2/h 2 - 2f'*

-2g*

]

-- 1

-2f."9"

"

Here h denotes step length and subscript n refers to value at the nth grid point. The tridiagonal system (25) was solved by initially assigning approximate values to f, f ' , and g and successively improving these values by using the Thomas algorithm (for details see Sherman (1990) page 185) iteratively. In each iteration, f was kept constant, and after obtaining the value of f ' , f was replaced by new values obtained by numerical integration of f ' .

5. Results and discussion The main results of this paper are shown in Figs. 2-10 for different values of rotation and permeability parameters s and K in the range 0 ~< s, K ~< 1.5. (i) Radial velocity profiles Fig. 2 shows that compared to rotation, the effect of permeability on radial velocity profiles is far more significant. In the absence of rotation, U / R ~- 1, depending on whether K ~ 1; furthermore,

154

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147 160 1.4 1.2 1 U 0.8 R

0.6 0.4 O.2

o O

K= 0

0.5

1

1.5 Z (a) s=O

2

2.5

2

2.5

3

1.4 1.2 1 LI

0.8

R

0.6 0.4

0 00

. 0.5

2 1

1.5 Z (b) s=1.5

Fig.2. Radial velocity profiles U/R asymptotically attains unity at Z ~- 2, thereby implying that for values of s and K in the range 0 ~< s ~< 1.5, the boundary layer is confined between the planes Z = 0 and Z -~ 2. (ii) Transverse velocity profiles The transverse velocity profiles shown in Fig. 3 are seen to be quite sensitive to variations in both rotation and permeability. Unlike radial velocity, the transverse velocity is found to decrease throughout the boundary layer with increase in permeability. (iii) Axial velocity on the bed An immediate result inferred from Fig. 4 is that for any value of permeability number K, the axial velocity on the bed increases with an increase in rotation. The effect of rotation is to throw the fluid near the bed radially outward; this in turn induces an axial flow into the porous bed. However, an interesting feature of the flow is that to each value of s there corresponds a definite non-zero value Ko of permeability number for which W (0) { 0 depending on whether K ~ Ko; this, in turn, implies the existence of a thin sublayer attached to the rotating bed in which the axial velocity is a directed upward although the potential flow far from the bed is moving downward. For different values of

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160 0.6 0.5 0.4

!=os\

0.3

0,2 'K~I 0,1 0

I

0

0.5

1

1.5 Z

2

2.5

3

(a) s = 0.5 1.6 1.4 1.2 1

V

0.8 0.6 0.4 0,2 0 0.5

0

1

1,5 Z (b) S=1.5

I

I

2

2.5

3

Fig. 3. Transverse velocity profiles.

0.4 S=l

W(O)

5=2

0.2

0

-0.2

-0.4 -0,6 0

I

I

I

I

I

I

I

0.2

0.4

0.6

0.8

1

1.2

1.4

K

Fig. 4. Normal velocity at the bed.

155

156

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160

s and K, the location Zo of the point in the boundary layer at which the axial velocity vanishes or changes direction is shown in Fig. 5. It is observed that corresponding to a given value of s, there exist two distinct non-zero values of K for the location where the axial velocity changes direction in the boundary layer. (iv) Radial and transverse slip velocity on the bed Fig. 6 shows that radial slip on the bed is insensitive to rotation. As expected, its value is greater for a bed of higher permeability. On the other hand, Fig. 7 shows that the transverse slip velocity on the bed increases with increase in rotation and decrease in permeability. Linear dependence of transverse slip velocity on s and K follows immediately from boundary condition (20). (v) Radial and transverse shear stress The responses of these two quantities to variation in rotation for different values of K is displayed in Fig. 8 and Fig. 9. The effect of rotation on transverse shear stress is seen to be far more

0.25 Zo s=1

0.2

0.15

0.1

0.05

0 0

0.2

0.4

0.6 K

0.8

1

1.2

Fig. 5. Location of point where the axial velocity changes direction.

1.6

K=2

f'(O) 1A

K=1.5

1.2 K=I

1

0.8 K = 0.5

0.6 0.4 0

I

0.3

0[6

019 s

Fig. 6. Radial slip velocity at the bed.

I

1.2

1.5

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160 1.6 g(O) 1.4 1.2 1

J

0.8 0.6 Q.4 0.2

o

o

0.3

0.6

0.9

1.2

1.5

S

Fig. 7. Transverse slip velocity at the bed.

2 f"(O) 1.5

K = O ~

1

KK =0.5

0.5 K=1.0 0 K= 1.5

-0.5 -1

K= 2.0

-1.5 0

L

L

I

I

0.3

0.6

0.9

1.2

1.5

S

Fig. 8. Radial shear stress.

o g'(O) -0.5

=' K = ' ~ -1.~

-2 0

i

1

I

I

0.3

0.6

0.9

1.2

S

Fig. 9. Transverse shear stress.

1.5

157

158

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147 160

~ 0

K=O I

0

0.5

1

1.5 Z (a) s=O

i

i

2

2.5

12 F(Z)

K =0.5 10 8 K :1.5 ~

6

K=i

4 2 0 0

0.5

1

1.5 Z (b) s=1,5

2

2.5

3

Fig. 10. Pressure function.

significant than its effect on radial shear stress. For a given value of s, there corresponds a value Ko of permeability close to unity for which the radial shear stress vanishes at all points on the bed. Unlike the transverse shear stress, which maintains the same sign with an increase in K, the radial shear stress is found to change sign during transition in the value of K through Ko. As expected, the radial shear stress is maximum in the case of an impermeable bed (K = 0). On the other hand, transverse shear stress (its absolute value) decreases with increases in rotation and permeability.

K.N. Mehta, K. Narasimha Rao / Fluid Dynamics Research 16 (1995) 147-160

159

(vi) Pressure distribution in the b o u n d a r y layer It has been observed that axial velocity may vanish in the b o u n d a r y layer region at a location depending on the values of s and K. This explains the relative placement of the plots of F for different values of K in the b o u n d a r y layer flow. Outside the b o u n d a r y layer but not far from the bed, the pressure function F is found to increase with permeability (Fig. 10); Eq. (7) would then imply that pressure decreases with an increase in permeability.

6. Conclusions A geophysical flow produced as a result of a column of an axisymmetric potential flow impinging on a rotating saturated porous bed of infinite extent is modelled as a b o u n d a r y value problem by selecting an appropriate slip b o u n d a r y condition. The response of the dynamic coupling between the surface b o u n d a r y layer and the Darcy flow separated by the permeable interface to variations in rotation and permeability is analyzed numerically by a combination of iteration and shooting methods. Some unexpected results, like vanishing of radial shear stress and axial velocity at all points on the bed, are the main findings of this paper. Subject to experimental validation, the theoretical predictions made in this study may prove useful in centrifuges and filtration processes.

Acknowledgements The authors express their grateful thanks to the referees for several comments and suggestions which proved very useful in revising the paper.

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Mehta, K.N. (1994) Unsteady flow past a naturally permeable bed, Encyclopedia of Fluid Mechanics, Supplement 3, Advances in Fluid Mechanics (Gulf Publishing Company) pp. 163-170. Mehta, K.N. and K. Narasimha Rao (1992) Development of transient flow in presence of slip at the boundaries, Modelling, Measurement and Control, B 43, 1-13. Mehta, K.N. and K. Narasimha Rao (1993) Plane flow impinging and flowing over a porous bed, J. Hydraulic Research, 31 pp. 563-573. Mehta, K.N. and K. Narasimha Rao (1994) Dynamic interaction between boundary layer flow and ground water flow separated by a porous bed, Int. J. Engn 9. Sci. 32, 654-652. Moore, F.K. (1956) Three-dimensional boundary layer theory, Advances in Appl. Mech. IV 159-228. Parter, S.V. (1982), On the swirling flow between rotating coaxial disks: a survey, Theory and application of singular perturbation, eds. W. Eckhaus and E.M. de Jager, Lecture Notes in Mathematics, Vol. 942, pp. 258-280. Saffman, P.G. (1971) On the boundary condition at the surface of a porous medium, Studies in Applied Math. 1, 93-101. Schlichting, H. (1979) Boundary Layer Theory (McGraw Hill). Sherman, F.S. (1990) Viscous Flow (McGraw Hill). Sparrow, E.M., G.S. Beavers, J.S. Chen and J.R. Lloyd (1973) Breakdown of the laminar flow regime in permeable-walled ducts, J. Appl. Mech. 10, 337 342. Stewartson, K. (1953) On the flow between two rotating coaxial disks, Proc. Cambr. Phil. Soc. 49, 333-341. Stuart, J.T. (1954) On the effects of uniform suction on the steady flow due to a rotating disk, Quart. J. Mech. Appl. Math. 7, 446-457. Taylor, G.I. (1971) A model for boundary condition of a porous material part I, J. Fluid Mech. 49, pp. 319-326. Wang, C.Y. (1979) Viscous flow between rotating discs with injection on the porous disc, Z A M P 30, 773-787. Zandbergan, P.J. and D. Dijkstra (1987) Von Karman swirling flows, Ann. Rev. Fluid Mech. 19, 465-491.