DEVELOPING LAMINAR FLOW IN THE ENTRANCE ...mecflu2.usuarios.rdc.puc-rio.br/Grad_Eng1707/Exercicios/...1518 C. NOUAR et al. Table 1. Main investigations on steady laminar developing

Pergamon Int. J. Engng Sci. Vol. 33, No. 10, pp. 1517-1534, 1995

Copyright ~) 1995 Elsevier Science Ltd 0020-7225(95)00016-X Printed in Great Britain. All rights reserved

0020-7225/95 $9.50 + 0.00

DEVELOPING LAMINAR FLOW IN THE ENTRANCE REGION OF ANNULI- -REVIEW AND EXTENSION

OF STANDARD RESOLUTION METHODS FOR THE HYDRODYNAMIC PROBLEM

C. NOUAR I.N.P.L-E.N.S.E.M-L.E.M.T.A., 2, avenue de la ForSt de Haye, B.P. 160, 54504 Vandoeuvre Cedex,

France

M. OULDROUIS and A. SALEM Laboratoire Cle M6canique des Fluides, Institut de Physique, U.S.T.H.B, Babezzouar, B.P. 32-El-Alia

Dar EI-Beida, Algeria

J. LEGRAND I.U.T de St-Nazaire, Laboratoire de G6nie des Proc6d6s, B.P. 420, 44606 St-Nazaire Cedex, France

A~t rae l - -Th is paper deals with the theoretical and numerical resolution of the laminar axial developing flow in annuli. In the first part of the article, a review of the different methods of the hydrodynamic problem resolution is given. We have then extended some of the methods performed for circular dacts to annular geometry. In particular we have developed a numerical solution of the complete Navier-Stokes equations, for studying the axial momentum diffusion effect on the flow structure. In the second part of this paper we have compared the results obtained with the different methods and analysed the hydrodynamics characteristics (velocity, pressure distribution and entrance length) as a fimetion of the annular flow parameters (radii ratio, Reynolds number and axial distance from the entrance).

1. INTRODUCTION

The development of axisymmetric steady laminar flow of liquid into an annular duct is one of the most widely studied problems of hydrodynamics. This is due to its technical importance such as in heat exchangers, axial flow turbomachinery, and in many manufacturing and processing industries.

As a fluid flows through an annular duct its velocity profile undergoes change from its initial entrance form to 1Lhat of a fully developed profile at an axial location far downstream from the entrance. The characteristics of the flow in this region can be determined by the resolution of momentum equations. For this purpose, a variety of methods have been employed. Most approximate ana]Lyses of the problem involve some form of Prandtl's boundary layer approximation, and uniform velocity profile at the entry of the annular space. This approach is valid when the Reynolds number is sufficiently high, for neglecting the axial momentum diffusion. Nevertheless, for some process industries, and specially food industry, the heat exchangers, has to heat or cool a highly viscous fluid. For this situation, Reynolds number is low to moderate, and therefore we have to consider the complete momentum equations.

The various methods used, in the case of boundary layer assumptions, may be placed into one of three categories: (i) integral method, it was presented by Chung and Astill [1]. The cross section of the annular space was considered to be composed of two regions: boundary layers developing near the walls and an inviscid fluid core. A third polynomial velocity profile distribution was assumed in the boundary layer and Bernoulli's equation was used in the core to determine the pressure distribution in the axial direction; (ii) linearization of the momentum equation, the non-linear inertia terms in the momentum equation were linearized and the resulting equation was solved analytically. Three types of linearization were proposed, the first

1517

1518 C. NOUAR et al.

Table 1. Main investigations on steady laminar developing flow in the entrance region of annuli

Investigator(s) ri/ro Type of resolution Main result

Murakawa [11] 0.636 analytical Murakawa [12] 0.636 analytical Sugino [3] 0.2; 0.5 analytical

0.83 Chang and Atabek [2] 0-1.0 analytical

Sparrow et al. [4] 0.001, 0.01, analytical 0.05, 0.1, 0.2

0.4, 0.8 Manohar [13] 0 .1-0 .8 numerical finite difference

Shah and Farnia [14] 0.05-0.955 Coney and EI-Shaarawi [8] Coney and El-Shaarawi [8] 0.05-0.955 Chung and Astill [1] EI-Shaarawi and Sarhan [9] Gupta and Garg [10] 0.001-0.8 Garg [15] 0.001-0.8 Soundelgekar [16] Terhmina and Mojtabi [17] 0.83, 0.71

0.57

numerical (finite difference)

numerical (finite difference) integral method

numerical (finite difference) numerical (finite difference) numerical (finite difference) numerical (finite difference) numerical (finite difference)

slight effect of N on Le slight effect of N on Le

Le and AP unaffected by N

for N >0.5 Le = 0.18(1 - N) 2

effect of percentage of approach of fully developed flow on Le

Le for various N

Le = Nt8(Rm - 1) 1"85 Re °'s

used by Chang and Atabek [2] based on Targ's assumption, and the second used by Sugino [3] which is an extension of Langhar's method, and finally Sparrow e t al. [4] and Sparrow and Lin [5] proposed another linearization which can be considered as a synthesis of the two above precedent techniques; (iii) finite difference method; initiated by Bodoia and Osterle [6], for flow between parallel plates. It has been extended to the annular duct by Coney and E1-Shaarawi [7, 8], EI-Shaarawi and Sarhan [9], and Gupta and Garg [10]. The outline of main

investigations is summarized in Table 1. The aims of this paper are to:

- - rev iew the different methods used for solving the flow development in the entrance

region of annuli. - - resolve the complete momentum equations, i.e. taking into account the axial momentum

diffusion and pressure gradient in the radial as well as axial direction. A similar work was reported in the literature in the case of straight channel or straight tube (Brandt and Gillis [18]; Wang and Longwell [19] for the straight channel and Friedman e t al. [20], Wagner [21], Vrentas e t al. [22], Pagliarini [23], Mehrotra and Patience [24], for the

straight tube). - -analyse the results obtained by the different methods, with and without Prandtl 's

approximation, for all the elements of the dynamical field (velocity, pressure, entrance

length).

2. G O V E R N I N G E Q U A T I O N S

We consider an incompressible viscous flow in an annular space between cylinders (Fig. 1). The dimensionless form of the conservation equations are:

Z-momentum

two coaxial

uov+ ov oe 1 [ l O__(Rov +o2v 1 v . . . . + - - (1)

O Z OR O Z R e L R O R \ O R ] O Z 2]

Laminar axial developing flow in annuli 1519

r° t v I - _ o

Fig. 1. Coordinate system.

Z

R-momentum

u O V + v O V OP 1 F 0 /1 0 R V \ 02V] O Z O-R= OR t-Ree[0-R[R0-R( ))+~-~-SJ (2)

aaz (RV) + ~ (RV) = O.

mass conservation equation:

The boundary conditions are:

Z - 0 ; R = N , or R = I ;

Z = 0 ; N < R < I ; U = I .

The continuity equation in the integral form is:

x 1 f R U d R = -N2). (1

U = V = 0

The equations were rendered dimensionless using the following reduced variables:

u r ri. P - Po. v z Uoro U = - - " R =--" N = - - , P = V = - - " Z = - ; Re =

U0 ro ' ro pu 2 ' Uo ' ro v

With the usual Prandtl boundary layer assumptions, equations (1) and (2) reduce to:

OU vOU dP 1 [1 0 (ROU]] u ~ + aR = ~*~L~ ~:J"

(3)

(4)

(5)

(6)

In the coming sections, we present the various methods employed to solve equations (3)-(6), and the one w]hich was used in the case of complete Navier-Stokes equations. Then, comparison of the results obtained by these different methods is made.

3. PRESENTATION OF THE VARIOUS METHODS

3.1 Methods with boundary layer assumptions 3.1.1 Analytical method. The difficulty in solving equations (3)-(6) arises from the non-

linearity of the inertia terms. Accordingly, various analytical approaches proposed in the literature involve linearization of inertia terms. The first proposed by Chang and Atabek [2],

E$ 33-10-d

1520 c. NOUAR et al.

introduced Targ's assumption which replaces U(OU/OZ)+V(OU/OR) by OU/OZ. After integration with respect to R from N to 1, the resulted equation is solved by Laplace transform. --The second linearization proposed by Sugino [3], extends the Langhar's method for the

circular tube to the annular space. The terms U(OU/OZ)+ V(OU/aR) and -(dP/dZ) are replaced respectively by (1/Re)/3Zu and -a/Re, where a and /3 are functions of Z only. After using, t = R/3, the solution is constructed as a linear combination of modified Bessel functions of the first and second kind.

~ T h e third approach proposed by Sparrow et al. [4] and Sparrow and Lin [5] can be seen as a synthesis of the two previous methods. In this approach, the momentum equation (6) is linearized as:

0U 1 1 O (R~__/U) = A(z) + Re R 0R (7)

where e(Z) is an undetermined function of Z which weights the mean velocity, and A(Z) is also an undetermined function which includes the pressure gradient as well as the residual of the inertia terms. This substitution is valid at the walls of the annulus and in the developed region. An auxiliary coordinate Z* may be defined as:

dZ = e dZ*. (8)

The velocity is then after decomposed as the sum of the fully developed profile U~(R) and a difference velocity U*(Z*, R). This latter is sought in the form:

U* = ~ K,g,(R)exp(-aZz *) (9) i = 1

when the function g,. is a linear combination of Bessel functions of the first and second kind. 3.1.2 Momentum integral approach. This technique was used by Chung and Astill [1], it

considers the growth of boundary-layer on the inner and outer walls of the annulus. Since the growth of boundary layers is asymmetric, each boundary layer must be treated separately.

The integral momentum equations are obtained after the following steps: --The momentum equation (6) is integrated from N to N + 8 i and 1 - 6o to 1, where N + 6i

and 1 - 60 are the edges of the inner and outer boundary layers, --The radial velocity component V is evaluated from the continuity equation. ~ T h e flow outside the boundary layers is assumed to be inviscid. - -A new radial coordinate Y is introduced such that Y = R - N for the inner boundary layer

and Y = 1 - R for the outer boundary layer. A third degree polynomial in Y is used for the velocity.

Finally a set of non-linear first order differential equations, is obtained and solved by a fourth-order Runge-Kutta algorithm.

3.1.3 Momentum energy integral approach. The Von Karman integral technique used above, assumes an inviscid fluid outside the boundary layers. This assumption may be valid near the entry, where the viscous effects are important only in the region near the walls, but becomes unrealistic in the important region near the fully developed flow. To avoid this assumption, Tiu and Bhattacharrya [25], determined the pressure drop term dP/dZ from the mechanical energy balance, which is obtained by multiplying equation (6) by U and integrating the result over the flow cross section to give:

de 2 [ l d f ; l f;uO__(ROU ] ] d Z - 1 - - - N 2 2d--Z RU3dR+R--e OR\ OR/dR" (10)

From now on, the analysis is similar to the momentum-integral method developed above, where a third degree polynomial is used for the velocity profile. On the other hand, the


momentum equation is integrated for the inner and outer boundary layers. After substituting the velocity profile and eliminating the pressure drop, a set of differential equations for the inner and the outer boundary layers are obtained. The solution of the differential equations is obtained numerically by the Adams-Moulton's predictor corrector algorithm.

3.1.4 Finite difference analysis. Equations (3)-(6) were solved by several workers (Coney [7, 8], EI-Shaarawi [9], Gupta [10], Soundelgekar [16], Terhmina and Mojtabi [17]), using an extension to the linearized, implicit finite difference technique of Bodoia [6]. The equation of continuity is rep]Laced by the following finite difference approximation:

Vi+l'j+l - Vi']+l .-}- Vi+l'j+l @ Vi'l+l q- Ui+l"j+l + U/'/+1 - Ui+l'l - ui'] = 0 AR 2Ri+½ 2 A Z

(11)

where R = N + (i - 1) AR; and Z = (j - 1) AZ. Terhmina and Mojtabi [17] suggest to approach the momentum equation by the following

finite difference:

Ui,1Ui,l~ . - Ui,j..~ Vii, U/+I, j+ 1 -I-- Ui+l, j4 AR- Ui-1,/+l - Ui-l,j

10/'+1 -- Pj}_ L[Ui+I , j+I q- U/+I , j - 2 U / d + 1 - 2Ui,i + U/_ I , j+ 1 + G- l , . / AZ Re [_ 2 ( A R ) 2

U/+I, j+ 1 q-- Ui+l, j - U/,j+ 1 - U/,j] + (12) 4R i AR J

Another similar ;approach was used by Coney and EI-Shaarawi [7, 8], Gupta [10], EI-Shaarawi [9] and Soundelgekar [16].

The integral representation of the continuity principle is reduced by the trapezoidal rule of numerical integration to the following equation:

Nmax 1 2 RiUi,j+x = - - ( I - N2) . ( 1 3 ) i=l 2 AR

Equations (12) and (13) give a set of (n + 1) simultaneous equations for the (n + 1) unknowns Ui,j+l at n mesh points of the radial position and Pj+I at the section j + 1 in terms of known values of Ui a and Pj at the section j. The set of algebraic equations is solved by using the sparse Gauss elimination method. The numerical procedure begins at the entrance section (Z = 0) and is stopped when the fully developed flow regime is obtained. From the values of U at section j + 1, the radial component of velocity at the mesh point is determined from the equation (11).

3.2 Resolution of' the complete Navier -S tokes equations

The method developed below constitutes an original resolution for developing axial flow in annuli. Although the boundary layer theory was the principle tool for solving the developing flow problem for the high Reynolds number, it involves replacing the elliptic Navier-Stokes equations by the parabolic form. It is well known that these assumptions are not valid for a range of small and moderate Reynolds numbers.

On eliminating the pressure gradient from equations (1)-(2), and introducing the dimensionless stream function ~b and the dimensionless vorticity to as:

1 oq 1 1 o$ oV oU U R O R ' V R O Z ' to OZ OR" (14)


Equations (1)-(3) and (14) lead to the following vorticity transport equation.

ozto 02to 0 R l ( 0 t o O V a t o 0 ~ ) t o 0 ~ ] 0Z2+~--~-~+~-~(R) = e [ ~ \ ~ - - ~ - ~ OR OZ/ + R 20ZJ"

From equation (14), the vorticity to is related to the stream function ~ by:

(15)

- - A t the outer wall; R = 1:

[W(N + 2 AR, Z) - 8 ~ ( N + AR, Z)] to(N, Z) = 2N(AR) 2 (19)

Boundary conditions for stream function qJ and vorticity to: The specification of to and 0 at the boundaries is the main difficulty in solving the boundary

value problem. - - A t the inner wall; R = N:

• =0. (17)

From equations (16) and (17), the vorticity to can be written as:

02~t -N to = OR--- 5 . (18)

By expanding in a Taylor series W(N + 2 AR, Z) and ~ ( N + AR, Z ) at the third order, it is possible to write:

The stream function qJ is calculated from the continuity equation (5), and equation (14)

u2(1, Z) = 0.5(1 - N2). (20)

In a similar manner as for R = N, the vorticity at the outer wall can be expressed as:

[ ~ ( 1 - 2 AR, Z ) - 8~P(1 - AR, Z ) - 7 ~ ( 1 , Z)I to(l, Z) = 2(AR)2 (21)

At the entrance, Z = 0, in order to avoid the nonphysical singularity associated with the discontinuity in U at the points (R = 1; Z = 0) and (R = N, Z = 0), a parabolic axial velocity profile was assumed at the vicinity of the walls.

U(R, O) = (aiR 2 + b i g + ct)Oo,

v ( g , o) = 0o,

U(R, O) = (a2R 2 + b2R + c2)Uo,

N ~- R -- N + AR (22a)

N + AR


using equations i',16) and (22). The second derivative (O2tIJ/0Z 2) is obtained by expanding in Taylor series, ~ ( 2 AZ, R) and W(AZ, R) at the third order.

al Oo -4 ~ A ~ cl Uo tit(R, 0) = ---~-- [/¢ - N 4] + [g3 - N3] + T [R2 -- N2];

w(R, 0) = I R 2 - (U + AR) 2] + qJ(U + AR);

q.t(R, 0)= aT-if°[1-R4]-~[1-R3]--~.0---~[1-R2];

with

N < R < _ N + A R

N + A R < - R < - I - A R

1 - A R < - R < - I

1 O2u~ to(R, 0) = - ( 2 a i R + bl)g/0 - R aZ 2 ; N ~ R ~ N + A R

1 02~ to(R, O)= R OZ 2' N + A R < - R < I - A R

1 02~ to(R, 0) = - (2a2R + b2)Uo - R OZ 2 ; 1 - A R < - R < - I

02~_~ 8tlJ(R, AZ) - W(R, 2 AZ) + 7W(R, 0) aZ 2 2 AZ 2

Far downstream from the entrance: Z--->o0 (Z = Z®): Z= corresponds to the axial position where the flow is fully developed. The axial velocity profile U®, at Z = Zoo, is given by

U~o = 2 (1 - R 2 + 2A log R) B

with

A (N 2 - 1) B -- 1 + N 2 - 2A. 2 In(N) '

The stream function us, and the vorticity to can be determined at Z = Zoo.

Ils(R,Z=)=[R-~B][1--RT+ 2,A(log(R) - ~)] + C

with

Numerical solution

For solving equations (15) and (16) with the appropriate boundary conditions, the following unsteady equations are considered:

o,','


coordinate transformation has to be applied to map the infinite region 0


Boundary conditions

The boundary conditions are of the second kind (the Neumann boundary conditions) specifying (OP/On)(R, Z), where n is normal to the boundary. The values of the pressure gradient are found from (1)-(2), and given below:

First an arbitrary pressure level is fixed at ~ = 1. (Z ~ o0)

P(R, 1) =0; N

1526 C. N O U A R et al.

The integration of the last equation between the entrance Z = 0, and any section at Z give:

pU 2 - 1 - N 2 1 ~ e RU2dR 1--~V2ff, e ~ 1 - N a-R N dZ. (32)

The integrals are calculated by the trapezoidal rule.

4. RESULTS AND DISCUSSION

4.1 Axial velocity development

Figure 2 shows the radial variation of axial component of the velocity at several axial positions, with radii ratio equal to 0.8. The profiles are determined by the finite difference method in the case of boundary layer analysis. They depend only on the parameter Z / R e . The same kind of profiles are obtained by the other methods based on boundary layer approximation.

The development of the axial velocity fields in the case of numerical resolution without boundary layer assumption is illustrated for different Reynolds numbers in Figs 3(a)-(c). These figures exhibit, for the axial positions close to the entry, a concavity and bulges. The velocity profiles have a local minimum at Rm (Rm is the radial position of the maximum of the axial velocity for fully developed flow) and a maxima near the walls. This effect was noted previously for flow in a straight channel by Brandt and Gillis [18], and by Wang and Longwell [19], and in a straight duct by Friedman et al. [20], Wagner [21], Vrentas et al. [22], Pagliarini [23], and Mehrotra and Patience [24].

Since we considered a continuous velocity profile at Z = 0 (equations 22a-c), contrary to most authors, and we obtained the bulges, we confirm that they are not due to the nonphysical singularity in U at the entrance section.

Abarbanel et al. [31] proved that the bulges do not represent a numerical effect arising from truncation errors. As a matter of fact, they appear in an exact analytical solution of Stokes flow problem in a quarter plane with appropriate boundary conditions so that it can be considered (near the entry) physically similar to the flow in the inlet region of a straight channel.

To visualize the phenomenon, see Table 2 and Fig. 4, we have calculated the difference A = Umax- Umin, where Umax and Umi n represent respectively the greatest value of the velocity profile, and the lowest value in the concavity, at various axial positions. Figure 4 indicates that

1.5o

1 .oo

U

0.50

o.oo 0.80 0.85 0 .90 0.95 1.00

R

Fig. 2. Development of axial velocity profiles (N = 0.8, determined by finite difference method, and based on boundary layer approximation: ( 0 ) Z = 10 -3 Re; (A) Z = 2 x 10 -3 Re; (11) Z = 5 x 10 -3 Re;

(*) Z = 2 × 10 -2 Re; (O) Z = 8.3 × 10 -2 Re.


1.50 1.50

1,00 1.00

U u

0.50 0.50

0.00 0.00 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00

R R

1.50

1.00

U

0.50

0.00 0.80 0.85 0.90 0.95 1.00

R

Fig. 3. Development of axial velocity profiles (N = 0.8): Resolution of complete momentum equations by finite diffeTence: (a) Re = 10, ( 0 ) Z = 0; (&) Z = 0.016; (11) Z = 0.033; (*) Z = 0.066; (O) Z = 0.15. (b) Re = 100, ( 0 ) Z = 0; (&) Z = 0.016; ( 1 ) Z = 0.033; (*) Z = 0.066; (O) Z = 0.1083; (I-q) Z = 0.245; (c) Re = 500, ( 0 ) Z = 0; (&) Z = 0.016; (l l) Z = 0.033; (*) Z = 0.083; (O) Z = 0.202; (1:]) Z = 0.973.

A increases with Reynolds number for the range 10-300, and decreases weakly from Re = 400 to 500. It also shows that A increases with Z at the two first sections, when Re > 140 and after decreases with Z until cancelled. Whereas for Re --- 140, A decreases from the first section. The corresponding radial position Rmax of /-/max moves to Rm when Z increases, and for a fixed position it approaches the wall when Re increases as indicated in Table 3.

Figure 5, where, A is plotted versus relative axial position Z/Le, for three axial positions Z is another representation of the evolution of A. It indicates that in the beginning A increases for Z/Le, and after decreses with Z/Le until cancelled. It can be assumed that A equals zero just at the entrance, and after increases until a maximum, followed by a decrease up to A = 0.

Comparison between axial velocity profiles obtained with and without boundary layer

Table 2. A x 102 at different axial positions and for various Reynolds numbers

z•e 10 20 30 40 50 60 80 100 120 140 160 250 300 400 500 0.016 4.4 5.2 5.9 6.4 6.8 7.2 7.5 7.7 8.0 8.2 8.87 8.67 8.76 8.69 8.46 0.032 4.2 1.3 2.3 3.2 4.0 4.7 5.9 6.7 7.3 7.8 9.25 9.06 9.30 9.48 9.55 0.048 0.0 0.0 0.0 0.35 0.94 1.6 2.8 3.9 5.4 5.6 7.90 7.75 8.17 8.62 8.90


0.1

A 0.01

t x ** ,

* ~: x *

t~

St

O

0 x

~t

gb

¢ t

0 . 0 0 1

0.01 . . . . . . 0/1 Z

Fig. 4. A against d imensionless axial pos i t ion Z fo r var ious Reyno lds numbers ( 6 ) Re = 10, (O ) Re=20, (I-7) Re=30, (A) Re=40, ((>) Re=50, (~-) Re=70, (+) Re=80, (x) Re= 140), (~)

Re = 250, (~t) Re = 500.

Table 3. Radial and axial positions of the bulges, for different Reynolds numbers

Z Re

20 100 500

0.016 0.833 0.826 0.820 0.032 0.853 0.833 0.826 0.048 - - 0.856 0.833

A

0.1 0 0 0o%° ~9o

rl 0 0 0

% []

t~

rl

o n 0

0

El

0.01 0

0.01 . . . . . . 0.~1 Z/Le

Fig. 5. A vs Z/Le for different axial positions: (O) Z = 0.016, ([3) Z = 0.032, (~) Z = 0.048.

a p p r o x i m a t i o n for the same R e y n o l d s n u m b e r , ind ica te tha t for the l a t t e r case the veloci ty

g rad ien t s c lose to the walls a re larger .

4.2 Friction fac tor

The m o m e n t u m t ransfe r at the walls can be cha rac t e r i zed by the fe loci ty g rad ien t s S o r by

the f r ic t ion fac tor f. T h e r e is a d i rec t r e l a t ion b e t w e e n S and f ; f = 2 - S / R e . The ve loc i ty

g rad ien t can be eas i ly e v a l u a t e d f rom the ve loc i ty profi les , and by using a second o r d e r

p rec i s ion d i f fe rent ia t ion , f is t hen d e t e r m i n e d f rom the a b o v e re la t ion .


1 0 -

, , t

3 t- O U 'E

,+_

"6 o o,"

1

(a )

x

~x N\NN \ ~ N = 0.5

~ ' ~ N X \ \ \ \ \ N = 0 . 8 ~ " ~ \ "

x X \ x

, , , , , , , , i . . . . . . . . i 1 0 1 0 0 Z 10 2

1 0 -

3 c

O

0 ID

t'v" 1

(b)

. . . . . . . . u , , , , , , , , i 1 0 1 0 0 Z . 1 0 2

1 ( C )

5

3 g

2 "6 0 rY"

, , ,. , , , , , , , , , ,,,,i 1 1 O 0

Z 1 0 2

J

(J & 0

rY

'" (d)

3"

, , , , , , , , u , , , , , , , , u

1 0 1 0 0 Re

Fig. 6. Friction factor ratio (f[f~) evaluated by resolution of complete momentum equations. (a) Against axial position (N = 0.8, Re = 100): ( ) inner wall, ( . . . . ) outer wall; (b) against axial position Z, for various Reynolds numbers ( N = 0.8): (O) R e = 2 0 , (A) Re = 100, (11) Re = 250, (~t) Re = 50t); (c) against axial position Z (Re = 100)/radius ratio effect: (O) N = 0.9; (V) N = 0.8; (U) N = 0 . 7 ; (.k) N = 0 . 6 ; (*) N = 0 . 5 ; (d) against Reynolds number for various axial positions:

(*) Z x 10 z = 3.33, (11) Z x 10 z = 5.00, (V) Z x 10 z = 6.67, (O) Z × 102 = 8.35.

Figure 6(a) shows the ratio o f f over f= as a function of the dimensionless axial position Z for two radii ratio N = 0.8 and 0.5 and for Re = 100. Three parts are observed in the evolution of the friction factor (i) sharp decrease close the entry, (ii) weak decrease where the ratio tends asymptotically to unity, as the flow approaches the fully developed state, (iii) constant value of 1 corresponding to the fully developed flow. Figure 6(a) also indicates that the ratio of friction factor f/f~ is larger on the outer wall than on the inner one. This difference increases with the decrease of the radii ratio, and is due to the dissymetry of the velocity profiles. Note that the velocity gradient is larger on the inner wall than on the outer one, but the ratio S/S= is larger on the outer wall.

The effect of tile Reynolds number on the ratio of friction factor is shown in Fig. 6(b), where f/f= is plotted against the axial position Z, for four values of Reynolds number. This figure indicates that the ratio f/f~ increases with Reynolds number, due to the decrease of the boundary layer thickness for a given axial position.

As the flow structure depends on the radii ratio, we have studied the effect of N on the ratio fir=. Figure 6(c) shows the evolution of the ratio of friction factor evaluated on the inner wall, against the axial position for five values of radii ratio N = 0. 5 . . . . . 0.9. For the same axial position the ratie of friction factor decreases with increase of N.


Table 4. Dimensionless entrance length (N=0.8) for various Reynolds number. (1) Finite difference method with axial diffusion; (2) finite difference method without axial diffusion;

(3) analytical method; (4) energy integral method

Re 8 10 15 20 30 40 50 60 70 80

Le(1) 0.1343 0.1366 0.1418 0.1465 0.1572 0.1686 0.1815 0.1956 0 . 2 1 0.2247 Le(2) 0.0136 0.017 0.0255 0.034 0.051 0 .068 0 .085 0 .102 0 .119 0.136 Le(3) 0.0144 0.018 0 .027 0 .036 0 .054 0 .072 0 .089 0 .107 0 .125 0.1428 Le(4) 0.013 0.0195 0.026 0 .039 0 .052 0 .065 0 .078 0.091 0.104

Re 90 100 120 140 160 200 250 300 400 500

Le(1) 0.237 0.2567 0.2903 0.3250 0.3599 0.4299 0.5112 0.5905 0.778 0.9545 Le(2) 0.153 0.17 0.204 0 .238 0.272 0.34 0.425 0.51 0.68 0.85 Le(3) 0.1606 0.1785 0.2142 0.25 0.285 0 .357 0 .446 0 .535 0 .714 0.8925 Le(4) 0.117 0.13 0.156 0 .182 0.208 0.26 0.325 0.39 0.052 0.65

Finally, for the purpose o f giving correla t ion for the ratio o f the friction factor, we have

represented in Fig. 6(d) f / f~ against Reynolds n u m b e r for different axial positions. This figure indicates, that for establishing correlat ions, we have to consider three regions for the Reynolds

number .

(a) 8 < Re < 35, and 10 -2 < Z < 6 x 10-2:

(b) 35 < Re < 100, and 1 0 - 2 < Z < 8 × 1 0 - 2

(c) Re > 100, and 10 -2 < Z < 12 × 1 0 - 2 :

(f / f~)i = 0.79 Re°°4Z-°73(1 - Rm) °'7

(f]f=)o = 1.26 Re°'°4Z-°73(1 - R m ) ° 9

(f/fao)i = 0.58 Re°'13Z-°'73(1 - Rm) °'7

(f / f~)o = 0.93 Re°13Z-° 73(1 - g m ) 0"9

(f/f=)i = 0.41 Re°21Z-°73(1 - R m ) °'7

(f/f®)o = 0.65 R e ° 2 1 Z - ° 7 3 ( 1 - Rm) °9.

W e can note that: (i) the influence o f radius ratio is smaller on the inner wall than on the outer ,

and is independent of Re; (ii) the decrease in the friction factor with axial distance has the same fo rm for all the considered range o f Re.

4.3 Entrance length

The ent rance length Le is defined as the dimensionless axial posit ion at which the max imum

velocity attains 99% of the max imum velocity for fully deve loped flow. For the case where the

axial m o m e n t u m diffusion is taking into account , Le was calculated for 20 values of Reynolds

n u m b e r ranging f rom 10 to 500, with radii ratio equal to 0.8. It was also calculated for five

values of radii ratio 0.9; 0.8; 0.7; 0.6; and 0.5 with Re = 100. These values are listed in Tables 4 and 5.

Figure 7 shows in logari thmic scale, the en t rance length against Reynolds number , obta ined f rom different numerical methods . It indicates that Le increases with Re. A t low Reynolds

Table 5. Dimensionless entrance length for various radii ratio

N 0.5 0.6 0.7 0.8 0.9 Le 1.324 0.854 0.505 0.260 0.100


Le 0.1

0 . 0 1

D

' ' ' , , , , , f , , , , , , , , i , , , , , , , , i 10 100 1000

Re

Fig. 7. Entrance length vs Reynolds number: ([]) finite difference method without axial diffusion, (O) analytical method, (1) energy integral method, (@) finite difference with axial diffusion.

numbers (Re < 200), the entrance length obtained by q-to method is larger (10 times at Re = 10) than those predicted by the boundary layer analysis. As the Reynolds increases, the difference between the entrance length obtained by ~b-o~, and boundary layer analysis decreases.

The effect of radii ratio, on the entry length is shown in Fig. 8, and Table 5. The entry length decreases with increase of N.

The methods based on boundary layer analysis give practically the same Le, except the method which uses the integral approach. This latter gives lower values.

All these results can be explained as: (i) the axial diffusion of the momentum causes the velocity to be spread downstream and the resulting entrance length is larger than those predicted by the boundary layer analysis, which considers only the convective transport of vorticity in the axial direction; (ii) the energy integral method introduces an additional approximate assumption in the velocity profiles, and that leads to a different values of Le than those predicted by the other methods based on boundary layer approximation.

For determini~Lg an analytical correlation between the entrance length Le and the parameters N and Re, we have assumed as Terhmina and Mojtabi [17], by analogy with the case of a flat plate, that the thickness of the outer (inner) boundary can be written as 8e = K Z - " Re-" where K, m and n are positives. It is obvious that 8e(Le)= 1 - R m. Finally Le can be written as:

Le

10-.

0.1

[ ]

0

0

0

[ ]

0

D

' ' ' '0~1 . . . . . . . i 1 - Rrn

Fig. 8. Entrance length vs 1 - Rm: (O) our results; (D) Gupta's results; (*) Coney's results.


Table 6. Different correlations for the entrance length

Finite difference with axial diffusion 8


~2

i

0 w , , , , , , , , i , , , , , , , , 1 1 , , , , , , , , , 1 , , , , , , i , , I

0 5 10 15 20 Z / R e

Fig. 10. Axial pressure drop: (11) finite difference method without axial diffusion, (O) analytical method, (*) energy integral method, (a) finite difference with axial diffusion.

methods based on boundary layer analysis give practically the same values, with a slight increment for the finite difference method.

5. CONCLUSION

The problem of the flow in the entrance region of an annular duct, based on boundary layer approximations was solved by various workers using different methods (integral method, linearization of Lhe inertia terms, and finite difference method). In this paper, we gave a particular consideration to the axial diffusion of momentum, in the range of Re = 10-500. The development of velocity profiles, friction factor as well as the determination of the entrance length and pressure drop were calculated by the above mentioned different methods. The main results can be summarized as follows:

(1) In the case of boundary layer approximations, linearization method, momentum energy integral method and finite difference method give practically the same distribution of axial velocity. The effect of axial diffusion of momentum is charac- terized by appearance of concavity in the velocity profiles. This concavity increases with increase of Z just near the entry, and after decreases with Z until cancelled. For a fixed axi~d position, the concavity increases with Re for the range 10 < Re < 300 and seems to decrease weakly for Re >300. The bulges approach the wall when Re increases. The representation of the concavity versus Z/Le, permits to concentrate the results ob~tained for different Reynolds numbers and axial positions.

(2) The friction factor values determined by boundary layer analysis are lower than those determined by a resolution of complete momentum equations. Different correlations have been proposed.

(3) The entrance length obtained by boundary layer analysis is shorter than those determined by the methods which considers the axial diffusion momentum. The difference decreases with increase of Reynolds number.

(4) The transverse pressure distribution is dependent upon axial position. As the value of Z increases the pressure variation with R approaches zero. Near the entry section, the pressure profiles show a concavity which is probably due to the radial velocity effect.

(5) As the velocity gradient on the walls evaluated by the resolution of complete momentu~T1 equations, are larger than those determined by the boundary layer analysis, therefore the axial pressure drop is also larger.


R E F E R E N C E S

[1] K. C. C H U N G and K. N. ASTILL, Trans. ASME J. Fluids Engng September, 558 (1976). [2] C. C H A N G and H. BOLENT A T A B E K , Z A M M 42, 425 (1962). [3] E. SUGINO, Bull. JSME 5, 651 (1962). [4] E. M. SPARROW, S. H. LIN and T. S. L U N D R E N , Phys. Fluids 7, 338 (1964). [5] E. M. S P A R R O W and S. H. LIN, J. Bas. Engng December, 827 (1964). [6] J. R. B O D O I A and J. F. OSTERLE, Appl. Sci. Res. Section A 10, 265 (1960). [7] J. E. R. CONEY and M. A. I. E L - S H A A R A W I , Int. J. Numer. Meth. Engng 9, 17 (1975). [8] J. E. R. CONEY and M. A. I. E L - S H A A R A W I , Nucl. Sci. Engng 57, 169 (1975). [9] M. A. I. E L - S H A A R A W I and A. S ARHAN, J. Heat Transfer Trans. ASME 48, 617 (1980).

[10] S. C. G U P T A and V. K. G A R G , Comp. Meth. Appl. Mech. Engng 28, 27 (1981). [11] K. M U R A K A W A , Int. J. Heat Mass Transfer 2, 240 (1960). [12] K. M U R A K A W A , Bull. JSME 3, 340 (1961). [13] R. M A N O H A R , Z A M M 4, 171 (1965). [14] V. L. S H A H and K. FARNIA, Computers Fluids 2, 285 (1974). [15] V. K. G A R G , 2nd Int. Conference, Venice, Italy, p. 439 (1981). [16] V. M. S O U N D E L G E K A R , Int. J. Engng Res. 11, 599 (1987). [17] O. T E R H M I N A and A. MOJTABI , Int. J. Heat Mass Transfer 31, 583 (1988). [18] A. B R A N D T and J. GILLIS, Phys. Fluids 9, 690 (1966). [19] U. L. W A N G and P. A. LONGWELL, A.I.Ch.E. Jl 10, 323 (1964). [20] M. F R I E D M A N , J. GILLIS and N. LIRON, Appl. Sci. Res. 19, 426 (1968). [21] M. H. W A G N E R , J. Fluid. Mech. 72, Pt 2, 257 (1975). [22] J. S. VRENTAS, J. L. D U D A and K. G. B A R G E R O N , A.LCh.E. Jl 12, 837 (1966). [23] G. PA G L IA RINI , Int. J. Heat Mass Transfer 32, 1037 (1989). [24] A. K. M E H R O T R A and G. S. PATIENCE, Can. J. Chem. Engng 68, 529 (1990). [25] C. TIU and S. B H A T F A C H A R Y Y A , Can. J. Chem. Engng 51, 47 (1973). [26] M. K A W A G U T I , J. Phys. Soc. Jap. 16, 2307 (1961). [27] Y. RIMON and S. I. CHENG, Phys. Fluids 12, 949 (1969). [28] H. M A S L I Y A H and N. EPSTEIN, J. Fluid Mech. 44, Pt 3, 493 (1970). [29] G. SHAVIT and Z. LAVAN, A I A A Paper No. 71-601, A1AA 4th Fluid and Plasma Dynamic Conference, Palo

Alto, CA (1971). [30] K. M I Y A K O D A , Jap. J. Geophys. 3, 75 (1962). [31] S. A B A R B A N E L , S. B E N N E T t , A. B R A N D T and J. GILLIS, Trans. ASME 37, 2 (1970).

(Revision received 16 May 1994; accepted 25 January 1995)

N O M E N C L A T U R E

f = friction factor N = radius ratio ( = ri/ro) p = pressure

P0 = pressure reference P = dimensionless pressure ( = (p - po)/pu 2) P = dimensionless mean pressure r = radial coordinate

R = dimensionless radial coordinate r~ = inner radius r o = outer radius

Re = Reynolds number ( = Uoro/V) Rm = radial position of the maximum velocity for

the fully developed flow S = dimensionless velocity gradient ( = OU/OR) u = axial velocity

Uo = mean velocity U = dimensionless axial velocity ( = u/Uo)

Ue = dimensionless free stream velocity U~ = fully developed velocity

v = radial velocity

V = dimensionless radial velocity ( = V/Uo) z = axial coordinate Z = dimensionless axial coordinate ( = z/ro)

Greek letters

8 = dimensionless boundary layer thickness ( = 8/ro)

A = difference between the greatest value of the velocity and the lowest in the concavity

p, = dynamic viscosity v = kinematic viscosity

= dimensionless stream function p = mass density to = dimensionless vorticity

Subscripts

i = inner cylinder o = outer cylinder

= fully developed flow

Documents

DEVELOPING LAMINAR FLOW IN THE ENTRANCE ...mecflu2.usuarios.rdc.puc-rio.br/Grad_Eng1707/Exercicios/...1518 C. NOUAR et al. Table 1. Main investigations on steady laminar developing