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Analytical solutions of laminar swirl decay in a straight pipe
Commun Nonlinear Sci Numer Simulat 17 (2012) 3235–3246
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Analytical solutions of laminar swirl decay in a straight pipe Shanshan Yao, Tiegang Fang ⇑ Department of Mechanical and Aerospace Engineering, North Carolina State University, 911 Oval Drive, Campus Box 7910, Raleigh, NC 27695, United States
a r t i c l e
i n f o
Article history: Received 4 March 2011 Received in revised form 22 November 2011 Accepted 30 November 2011 Available online 13 December 2011 Keywords: Laminar swirl flow Pipe flow Swirl nozzle Analytical solutions Developed flow
a b s t r a c t In this work, the laminar swirl flow in a straight pipe is revisited and solved analytically by using prescribed axial flow velocity profiles. Based on two axial velocity profiles, namely a slug flow and a developed parabolic velocity profiles, the swirl velocity equation is solved by the separation of variable technique for a rather general inlet swirl velocity distribution, which includes a forced vortex in the core and a free vortex near the wall. The solutions are expressed by the Bessel function for the slug flow and by the generalized Laguerre function for the developed parabolic velocity. Numerical examples are calculated and plotted for different combinations of influential parameters. The effects of the Reynolds number, the pipe axial distance, and the inlet swirl profiles on the swirl velocity distribution and the swirl decay are analyzed. The current results offer analytical equations to estimate the decay rate and the outlet swirl intensity and velocity distribution for the design of swirl flow devices. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Swirl flows have many important engineering applications such as the cyclone for separation of solid, liquid, and gas, swirl atomizers, swirl combustion devices, heat transfer enhancement, and others [1,2]. In most of these devices, swirl flow transport in a straight pipe is the most commonly used. Due to the wall effects inside the pipe, the swirl intensity decays as it moves downstream. Meanwhile, most of the applications require sufficiently large swirl intensity to maintain the performance of the devices and reduction of swirl decay becomes important. Therefore, the understanding and controlling the swirl transport in a pipe is of great importance in the design of these devices. The study of swirl decay in a pipe has a relatively long history. Some early works on the swirling flows related to a swirl atomizer were carried out with analytical techniques [3,4], where the boundary layer problem was solved for flow in a converging nozzle of a swirl atomizer. Talbot might be the first one who solved the laminar swirl flow in a pipe [5]. An early work on turbulent swirl decay in a pipe was performed by Kreith and Sonju [6], where an approximation solution was presented for a developed axial turbulent flow. The results were compared with experiments and good agreement was obtained in the region less than 20 diameters of the inlet. Later, theoretical analyses were conducted on laminar swirl flows in a pipe following the work of Talbot. Kiya et al. [7] studied the developing swirl flow in a pipe. The governing equations were solved numerically using finite difference method. A solid-body swirl was used in the inlet of the pipe. The flow development in the entrance region was solved. Singh et al. [8] analyzed the swirl flow development in the near inlet region based on boundary layer assumption and the problem was solved using similarity transformation for the swirl velocity near the wall. Series solutions for the same governing equations used in Ref. [8] were solved by series expansion and the wall shear stresses were calculated for blood swirl transport in a straight tube [9]. An analytical solution was presented by Reader-Harris [10] for a developed turbulent swirling pipe flow using the separation of variable technique and the solution was expressed by the Bessel functions. The swirl decay rate was given by a simple equation as a function of the distance along the pipe. An ⇑ Corresponding author. E-mail address: [email protected] (T. Fang). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.11.038
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analytical approach was used by Yu and Kitoh [11] to derive the swirl decay behavior for a straight pipe based on the conservation of angular momentum flux. Approximate solution was derived and good agreement with experiments was observed. One important finding is that the swirl decay in a straight pipe does not always following the exponential law and the decaying rate can depend on the downstream pipe wall drag coefficient, the flow Reynolds number, and the swirl intensity. Based on boundary layer assumption, the developing turbulent swirling flow through a fixed pipe was investigated numerically and analytically [12]. Recently, an integral analysis was performed for the entrance region based on boundary layer assumptions and the swirl decay rate was developed by Maddahian et al. [13]. Their results showed good agreement with the available experimental data. Ayinde [14] conducted numerical calculations on the laminar swirl flow in a pipe with a three-dimensional computational fluid dynamics (CFD) code and a swirl decay rate formula was proposed accounting the effects of the flow Reynolds number, the pipe length, inlet swirl intensity, and the inlet swirl velocity profile. The author showed the effects of swirl on the axial velocity distribution as the swirl transports along the pipe, however, the assumed inlet developed axial velocity can be recovered at the exit of the pipe. The velocity profiles between the inlet and the outlet falls between the slug flow profile and the fully developed parabolic velocity profiles. Based on the above discussion, although the laminar pipe flow has received significant attention, analytical solutions to this problem are not addressed sufficiently in the literature. In this work, the laminar swirl pipe flow is revisited analytically with exact solutions. In order to make the governing equations solvable exactly, it is assumed the axial velocity profile is developed and does not change along the pipe. The full swirl velocity equation will be solved for a slug axial flow and approximated swirl velocity equation will be solved for both slug flow and the developed parabolic velocity profiles. These solutions are presented explicitly and the swirl decay can be easily computed based on the results. In addition, because for the entrance region of a laminar swirl flow, the axial velocity profile would transition from a slug flow to a developed parabolic flow, the current results can provide estimated bounds for the swirl velocity and the swirl decay in the pipe. 2. Mathematical formulation and analytical solutions In this paper, we consider a laminar swirling flow in a straight pipe with a given swirling velocity profile at the inlet of the pipe. For incompressible fluids without body force and based on the axisymmetric flow assumption, the three-dimensional steady state Navier–Stokes (NS) equations in cylindrical coordinates read [15]
1 @ @uz ðrur Þ þ ¼ 0; r @r @z
ð1Þ
! @ur @ur u2h 1 @p @ 2 ur 1 @ur @ 2 ur ur þm þ þ uz ¼ þ 2 2 ; ur r @r @r @z r q @r @r 2 @z r ! 2 2 @uh @uh ur uh @ uh 1 @uh @ uh uh ur þ þ uz þ ¼m þ 2 2 ; r @r @r @z r @r 2 @z r ! 2 2 @uz @uz 1 @p @ uz 1 @uz @ uz ur þ þm þ uz ¼ þ 2 ; r @r @r @z q @z @r 2 @z
ð2aÞ ð2bÞ ð2cÞ
*
where the velocity vector is V ¼ ður ; uh ; uz Þ; m is the kinetic viscosity, p is the fluid pressure, and q is the density of the fluid. The three components in the cylindrical coordinates are r, h, and z. We assume the axial flow is developed and the axial flow velocity is specified as a function of r only, namely uz = uz(r). In addition, to solve the equation analytically using separation of 2 variables, we assume @@zu2h is small for a general axial flow uz = uz(r) at the beginning. But for a slug flow, we will consider all the terms in the swirl velocity later. Based on this assumption, it is obtained ur = 0 from the continuity equation and the governing equation for uh become
! @uh @ 2 uh 1 @uh uh uz ðrÞ þ ¼m 2 ; r @r @z @r 2 r
ð3Þ
with boundary conditions as
uh ðz; 0Þ ¼ 0;
uh ðz; aÞ ¼ 0;
uh ð0; rÞ ¼ uh;i ðrÞ;
ð4a; 4b; 4cÞ
where a is the pipe radius, and uh,i(r) is the inlet swirl velocity and in general, it is given as [14]
8 r < rt ; < uh;i;max rrt ; h i uh;i ðrÞ ¼ ar : uh;i;max rrt ar ; r P rt ; t
ð5Þ
where rt is the radial location for the swirl flow transitions from a forced vortex to a free vortex. The flow configuration is illustrated in Fig. 1. Before solving the swirling velocity equation, we define some dimensionless variables and velocities as
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Fig. 1. A schematic of the flow configuration.
z Z¼ ; a
r R¼ ; a
UðRÞ ¼
uz ; Ur
WðZ; RÞ ¼
uh ; Ur
ð6a; 6b; 6c; 6dÞ
where Ur is the reference velocity and it is the average axial velocity in the pipe. The governing equation (3) becomes
ReU
@W @ 2 W @ W ¼ ; þ @Z @R2 @R R
ð7Þ
with the boundary conditions as
WðZ; 0Þ ¼ 0;
WðZ; 1Þ ¼ 0;
Wð0; RÞ ¼
uh;i ðRÞ ; Ur
ð8Þ
where Re is the Reynolds number defined as Re ¼ aUm r . Since the swirling flow equation is linear, without of the loss of generality, we can set
8 < RRt ; R < Rt ; Wð0; RÞ ¼ R h 1R i : t ; R P Rt : R 1Rt
ð9Þ
To investigate the decay of the swirl intensity, a swirl number is defined as [13,16,17]
Ra S¼
uh uz r2 dr Ra ¼ SðzÞ: a 0 u2z rdr 0
ð10Þ
Substituting the dimensionless functions, the swirl number can be rewritten as
SðZÞ ¼
uh;i;max Ur
R1 0
WðZ; RÞUðRÞR2 dR : R1 UðRÞ2 RdR 0
ð11Þ
Then ratio of the swirl number at Z to the inlet swirl number is given by
R1 WðZ; RÞUðRÞR2 dR SðZÞ : ¼ R01 Sð0Þ Wð0; RÞUðRÞR2 dR
ð12Þ
0
d Based on Eq. (12), the swirl decay rate along the axial direction can be given as dZ
SðZÞ Sð0Þ
. Since the swirling velocity equation is
a linear partial differential equation, it can be solved by a separation of variable technique. Defining W(Z, R) = F(Z)G(R) and substituting it into Eq. (7) yield
_ ReUðRÞGðRÞFðzÞ ¼ FðZÞG00 ðRÞ þ FðZÞ
0 GðRÞ R
ð13Þ
Where a ‘‘dot’’ denotes a derivative with respect to Z, while a ‘‘prime’’ indicates a derivative with respect to R. By dividing both sides by F(Z)G(R) and separating variables, it is obtained
h i0 G00 ðRÞ þ GðRÞ _ R ReFðZÞ ¼ ¼ k2 FðZÞ UðRÞGðRÞ
ð14Þ
where k2 is a constant (called eigenvalue) to be determined later and k is a positive real number. The solution of F(Z) can be obtained as k2
FðZÞ ¼ e Re Z :
ð15Þ
The radial function G(R) becomes
R2 G00 þ RG0 þ ½k2 UðRÞR2 1G ¼ 0;
ð16Þ
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with boundary conditions
Gð0Þ ¼ 0 and Gð1Þ ¼ 0:
ð17a; 17bÞ
For a given axial velocity distribution, the solution of Eq. (16) can be obtained. In the current study, two typical axial flow profiles are used, namely the uniform slug flow and the developed parabolic velocity profile. Three cases are discussed as follows. 2.1 Case A: Slug flow with uniform velocity For a slug flow, U(R) = 1, Eq. (16) becomes
R2 G00 þ RG0 þ ½k2 R2 1G ¼ 0:
ð18Þ
The solution of Eq. (18) satisfying the boundary condition at R = 0 reads
GðRÞ ¼ Bessel Jð1; kRÞ:
ð19Þ
where Bessel J(a, x) is the Bessel function of the first kind [18]. The values of k, also called eigen-value, are determined by the boundary condition at the wall with R = 1,
Bessel Jð1; kn Þ ¼ 0:
ð20Þ
Based on the properties of the Bessel functions, there are an infinite number of eigen-values for this problem. The related solution of Eq. (19) for each given eigenvalue is called the eigen functions for this problem. Based on the Sturm–Liouville R1 theorem [19], the eigen-functions are orthogonal for any two given values of kn, namely 0 Bessel Jð1; km RÞ Bessel Jð1; kn RÞRdR ¼ 0 for m – n. The general solution of W(Z, R) reads
WðZ; RÞ ¼
1 X
k2 n
C n eRe Z Bessel Jð1; kn RÞ
ð21Þ
n¼1
Based on the orthogonality of the eigen-functions, the coefficient Cn is given by
R1 Cn ¼
0
Wð0; RÞBessel Jð1; kn RÞRdR : R1 Bessel Jð1; kn RÞ2 RdR 0
ð22Þ
2.2. Case B: Developed parabolic velocity profile For a developed parabolic velocity profile, uz(r) = 2Ur(1 R2). The governing equation for G(R) becomes
R2 G00 þ RG0 þ ½2k2 ð1 R2 ÞR2 1G ¼ 0;
ð23Þ
with the same boundary conditions as Eqs. (17a, 17b). The solution for Eq. (23) is
pffiffiffi Laguerre L 2pk ffiffi2 ; 1; 2kR2
pffi kR
2
e
2
GðRÞ ¼
R
ð24Þ
;
where Laguerre L(a, b, z) is called the generalized Laguerre function [18], and the boundary condition at R = 0 is satisfied automatically. By setting G(1) = 0, the eigen-values are given by
Laguerre L
pffiffiffi kn pffiffiffi ; 1; 2kn 2 2
¼ 0:
ð25Þ
The solution for the swirling flow is given by 2
WðZ; RÞ ¼
1 X n¼1
Cne
k2
Ren Z
knpRffi
e
2
Laguerre L
kpnffiffi ; 1; 2 2
R
pffiffiffi 2kn R2 :
ð26Þ
Based on the Sturm–Liouville theorem [19], the eigen-functions are orthogonal for any two given values of kn, namely km R2 k R2 pffiffi pffiffi pffi n pffi R 1 e 2 Laguerre L 2kpnffi2;1; 2kn R2 e 2 Laguerre L 2kpmffi2;1; 2km R2 Rð1 R2 ÞdR ¼ 0 for m – n. The coefficients in Eq. (26) can be obtained 0 R R based on the inlet swirling flow profile as
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R1 Cn ¼
0
k R2 p n 2 Laguerre L
ffi
e
ffi
kn p ;1; 2 2
pffiffi 2kn R2
Wð0; RÞ Rð1 R2 ÞdR R : 2 32 2 k R pffiffi pffi n kn 2 2 p ffi e Laguerre L ;1; 2 k R n R1 6 2 2 7 2 5 Rð1 R ÞdR 0 4 R
ð27Þ
2.3. Case C: Slug flow including axial swirl diffusion 2
In the previous derivation, we assume @@zu2h is small compared to the convection term. However, for the slug flow, we can include the axial diffusion term due to the simplicity of the velocity and the dimensionless swirl velocity equation becomes,
Re
@W @ 2 W @ W @2W þ : ¼ þ 2 @Z @R R @R @Z 2
ð28Þ
Following the previous procedure, we can separate the variables as
h i0 00 GðRÞ G ðRÞ þ _ € R ReFðZÞ FðzÞ ¼ ¼ k2n : FðZÞ GðRÞ
ð29Þ
We obtain the equation for the axial part,
F€ ReF_ k2n F ¼ 0:
ð30Þ
The solution reads
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
Re
FðZÞ ¼ e
4k þRe n 2
Z
ð31Þ
; 2
kn which can reduce to Eq. (12) for large Re, namely the slug flow solution without the axial diffusion term. However, Re is al pffiffiffiffiffiffiffiffiffiffiffiffiffi Re 4k2 þRe2 2 n for k > 1, which could lead to a faster swirl decay along the pipe for Case A than Case C as ways greater than n 2
shown in the result section. G(R) remains the same and all the eigen-functions and eigen-values keep the same as Case A. The general solution is given by
WðZ; RÞ ¼
1 X
Re
Cne
pffiffiffiffiffiffiffiffiffiffi ffi 2 2 4k þRe n 2
Z
Bessel Jð1; kn RÞ;
ð32Þ
n¼1
where the constant coefficients are given by Eq. (22). In order to obtain the swirling velocity distribution, the eigen-values for the two given axial velocity profiles must be solved first. All the presented results in this work were obtained using Mathematica 6.0 software. The eigen-values were calculated for both kinds of eigen-functions. In this work, n = 50 was used for all the calculations. Some examples of the eigenvalues, namely the first 30 eigen-values, are listed in Tables 1 and 2 for the Bessel function and the generalized Laguerre function, respectively. Before results are plotted and discussed, further analytical derivations are presented. Due to the simplicity of the Bessel R1 function, its integration can be expressed explicitly as 0 Bessel Jð1; kn RÞ2 RdR ¼ 12 Bessel Jð2; kn Þ2 . With the given inlet swirl flow profile (9), we can obtain
Z
( Rt Rt BesselJð0;Rt kn Þ BesselJð0;kn Þ BesselJð2;Rt kn Þ þ kn kn 1 Rt h i) 3 5 1 2 kn HypergeometricPFQ 2 ; 2; 2 ; 4 kn kn R3t HypergeometricPFQ 32 ; 2; 52 ; 14 k2n R2t ; 6
1
Wð0;RÞBesselJð1;kn RÞRdR ¼
0
Table 1 The first 30 eigen-values for the Bessel function of the first kind. n
kn
n
kn
n
kn
n
kn
n
kn
1 2 3 4 5 6
3.8317 7.0156 10.1735 13.3237 16.4706 19.6159
7 8 9 10 11 12
22.7601 25.9037 29.0468 32.1897 35.3323 38.4748
13 14 15 16 17 18
41.6171 44.7593 47.9015 51.0435 54.1856 57.3275
19 20 21 22 23 24
60.4695 63.6114 66.7532 69.8951 73.0369 76.1787
25 26 27 28 29 30
79.3205 82.4623 85.604 88.7458 91.8875 95.0292
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Table 2 The first 30 eigen-values for the generalized Laguerre function. n
kn
n
kn
n
kn
n
kn
n
kn
1 2 3 4 5 6
3.2697 6.1147 8.9485 11.7794 14.6092 17.4386
7 8 9 10 11 12
20.2676 23.0965 25.9252 28.7539 31.5826 34.4111
13 14 15 16 17 18
37.2397 40.0683 42.8968 45.7253 48.5538 51.3823
19 20 21 22 23 24
54.2108 57.0392 59.8677 62.6962 65.5246 68.3531
25 26 27 28 29 30
71.1816 74.0100 76.8385 79.6669 82.4954 85.3238
where Hypergeometric PFQ[(a), (b, c), x] is the generalized hypergeometric function. Therefore, the constants can be given explicitly as Rt kn
Cn ¼
Rt Bessel Jð2; Rt kn Þ þ 1R t
Bessel Jð0;Rt kn ÞBessel Jð0;kn Þ kn
kn Hypergeometric PFQ ½ð32Þ;ð2;52Þ;14k2n kn R3t Hypergeometric PFQ ½ð32Þ;ð2;52Þ;14k2n R2t
6
:
1 Bessel Jð2; kn Þ2 2
ð33Þ
The above equation gives the coefficients for both Case A and Case C. Then the swirl number can be given for the slug flow as R1 WðZ;RÞR2 dR u 0 SðZÞ ¼ h;i;max . For Case A with the given inlet swirl velocity, we obtain 1 Ur 2
Z 1 n 1 1 k2 2uh;i;max X 2uh;i;max X C n eRe Z Bessel Jð2; kn Þ n SðZÞ ¼ C n e Re Z Bessel Jð1; kn RÞR2 dR ¼ : kn Ur Ur 0 n¼1 n¼1 k2
ð34Þ
For Case C, it is obtained Re
SðZÞ ¼
1 X Cn e
2uh;i;max Ur
pffiffiffiffiffiffiffiffiffiffi ffi 2 2 4k þRe n 2
n¼1
Z Bessel Jð2; kn Þ : kn
ð35Þ
The swirl number ratio is then given for Case A by
SðZÞ ¼ Sð0Þ
k2 nZ Re Bessel Jð2;kn Þ
P1
P1
Cn e
n¼1
P1
n¼1
kn C n Bessel Jð2;kn Þ kn
Cn e
k2 nZ Re Bessel Jð2;kn Þ
n¼1
¼
R1 0
kn
P1 ¼
Wð0; RÞR2 dR
Cn e
k2 nZ Re Bessel Jð2;kn Þ
n¼1
kn
Rt =6 þ R2t =6 R3t =12
ð36Þ
and for Case C by
SðZÞ ¼ Sð0Þ
P1
n¼1
Re
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
4k þRe n Z 2 Bessel Jð2;kn Þ
Cn e
R1 0
kn
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
Cn e
4k þRe n Z 2 Bessel Jð2;kn Þ
n¼1
¼
Wð0; RÞR2 dR
Re
P1
kn
Rt =6 þ R2t =6 R3t =12
ð37Þ
:
For the parabolic velocity profiles, the swirl number becomes
SðZÞ ¼
uh;i;max Ur
P1
n¼1 C n e
k2 Ren Z
R1
k R2 p n 2 Laguerre L
ffi
e
0
ffi
kn p ;1; 2 2
Z 1e 1 k2 3 uh;i;max X n C n e Re Z 2 U r n¼1 0
2ð1 R2 ÞR2 dR
4ð1 R2 Þ2 RdR pffiffiffi ffi Laguerre L kpnffiffi ; 1; 2kn R2
2 knpR 2
¼
pffiffi 2kn R2
R
R1 0
2 2
R
2ð1 R2 ÞR2 dR:
ð38Þ
Due to the complexity of the generalized Laguerre function, the integration has to be evaluated numerically. The ratio of the swirl number reads
SðZÞ ¼ Sð0Þ
P1
n¼1 C n e
R1 0
e
k R2 p n 2 Laguerre L
R1 0
P1 ¼
k2
Ren Z
ffi
R 2
ffi
kn p ;1; 2 2
pffiffi 2kn R2
2ð1 R2 ÞR2 dR
2
Wð0; RÞR 2ð1 R ÞdR pffiffi ffi kn pffi ;1; 2kn R2 2 2
k R2 p n 2 Laguerre L e
R1 2ð1 R2 ÞR2 dR 0 R : 2 3 4 1 R 7 þ 7R 8R 3R þ 2R t t t t t 30
k2
Ren Z n¼1 C n e
ð39Þ
S. Yao, T. Fang / Commun Nonlinear Sci Numer Simulat 17 (2012) 3235–3246
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Fig. 2. Comparison of the current analytical solution with the direct numerical simulation of the PDE for Case B under given conditions.
Fig. 3. Swirl velocity profiles of Case A for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
3. Results and discussion In this section, several numerical examples are presented and the effects of different parameters on the swirl velocity profiles and the swirl decay along the axial direction are analyzed. Before the current analytical solutions were applied,
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Fig. 4. Swirl velocity profiles of Case B for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
Fig. 5. Swirl velocity profiles of Case C for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
validation was carried out to show the correctness of these solutions. Comparison is made in Fig. 2 for given values of control parameters using the current analytical series solutions and direct numerical calculation (using Mathematica) of Eq. (7) for Case B. It is seen that the results are in good agreement. The swirl velocity profiles for the three cases are illustrated in Figs. 3–5 respectively for different axial locations and the flow Reynolds numbers. In each figure, the top plot shows the effects of
S. Yao, T. Fang / Commun Nonlinear Sci Numer Simulat 17 (2012) 3235–3246
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the axial location from the inlet of the pipe on the swirl velocity distribution, while the effects of the Reynolds numbers are shown in the bottom plot. It is seen that for a given inlet swirl velocity distribution, namely Rt = 0.9 for the current examples, the magnitude of the swirl velocity becomes smaller with the increase in Z. Also the swirl velocity decay starts from the wall and penetrate into the center of the pipe with increasing the axial distance in the pipe. The effects of the flow Reynolds numbers on the swirl velocity, however, are quite different. The swirl velocity decays faster for a smaller Reynolds number when keeping the distance from the inlet the same. This is consistent with the physical observation. For the current problem, a higher value of Reynolds number implies a higher average axial velocity if the pipe diameter and the fluid are kept the same. This higher axial velocity transports the swirl motion downstream faster with less time for the wall to dampen the swirl velocity. A close scrutiny of the swirl velocity profiles reveals that there do exist some differences for the three cases. To further illustrate the differences among the three cases investigated in this paper, the swirl velocity profiles are compared for Cases A, B, and C with different combinations of Re and Z, shown in Fig. 6. There are four groups of parameters in the plots, including small Z and small Re, small Z and large Re, large Z and small Re, and large Z and large Re. All the plots are based on the same inlet swirl profile with Rt = 0.9. For a relatively small Re, the decay of the swirl velocity is very fast. The differences among the three cases are obvious. For Re = 10, the swirl velocities in Cases A and C decay faster than Case B in general. However, for a short distance from the inlet, Case B has a faster decay near the pipe wall in the near inlet region (Z = 1). This can be explained by the axial velocity distribution. For both Case A and Case C, the axial velocity near the wall is higher than that of Case B, say the parabolic velocity profile. This faster axial motion results in a slower decay of the swirl motion for a given distance from the inlet. In addition, for a given Re, Case A always decays faster than Case C as discussed before. The results for a high Re, however, are very different. With a higher Re, Case C and Case A give almost the same results and the curves are overlapped in the plot. Case B has a faster decay for the given values of Z due to a lower axial velocity in the near wall region. The swirl decay along the pipe, given by SðZÞ , is shown in Fig. 7 for Cases A and C for three values of the Reynolds numbers. Sð0Þ The results in Fig. 7 are consistent with the findings in Fig. 6. Case A leads to a faster swirl velocity decay compared with Case C. With the increase of the Reynolds numbers (Re > 100), the difference becomes very small. For Re < 10, the swirl velocity can be damped out within the distance of two pipe diameters. In order to reduce the decay of the swirl flow, a faster axial velocity and a large pipe or a less viscous fluid can help in this regard with increased flow Reynolds number. The decay difference between Case A and Case C can also be explained based on the governing equations and the flow configuration. For Case A, the axial swirl motion transport is only determined by the convective motion due to the axial flow velocity; while for Case C there is an extra axial diffusion contribution of swirl motion besides the convective swirl transport downstream along the pipe. A comparison of the swirl decay for Cases B and C is shown in Fig. 8. It is found that Case B provides a different swirl behavior. For a small value of Re, Case B decays faster than Case C, while for a relatively large Re, Case B decays slower than
Fig. 6. Comparison of the three cases for different combinations of the pipe location and the Reynolds number for Rt = 0.9.
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Fig. 7. The swirl decay as a function of the pipe locations for the three cases under different values of the Reynolds numbers for Rt = 0.9.
Fig. 8. The swirl decay as a function of the axial locations for Cases B and C under different values of the Reynolds numbers for Rt = 0.9.
Fig. 9. The swirl number ratio
SðReÞ Sð0Þ
as a function of the Reynolds number of Case C at different pipe locations for Rt = 0.9.
Case C (Re = 10 and Re = 100 in Fig. 8) for a large enough distance from the inlet. Near the inlet of the pipe, there exist a crossover between Cases B and C for a certain Re, which indicates that near the inlet Case B decays faster than Case C, as shown in the insert of Fig. 8. Based on the current comparisons among the three cases, it is concluded that the difference between Cases A and C becomes negligible for a sufficiently large Reynolds numbers. For small Reynolds numbers, both Cases A and B give overestimated swirl decay by neglecting the axial diffusion term, which can be comparable to the convective term. Therefore, for relatively large Re, Cases A and B provide good approximations for swirl velocity and its decay. In real physical applications, the velocity profiles in the pipe are not always developed and the interaction of the swirl with axial motion and radial motion of the fluids can affect the axial velocity profiles. However, the shape of axial velocity profiles is generally between the slug flow and the parabolic profile. Thus, Cases A and B can offer analytical upper and lower limits of swirl decay along the pipe.
S. Yao, T. Fang / Commun Nonlinear Sci Numer Simulat 17 (2012) 3235–3246
3245
Fig. 10. The swirl decay as a function of the pipe locations for Case C under different values of the Reynolds numbers for Rt = 0.25, 0.50, 0.75 and 0.90.
To further illustrate the effects of the Reynolds numbers on the swirl decay, SðReÞ is plotted in Fig. 9 for different axial disSð0Þ tances from the inlet. It is confirmed that increasing the Reynolds number reduces the swirl decay and increasing distance from the inlet enhances the decay. This has important applications in the design of the swirl driven devices. For example, in the design of the swirl atomizer, in order to form a conical liquid sheet at the nozzle exit, sufficiently large swirl number should be maintained. For a given nozzle with designed diameter and length, increasing the flow Reynolds number is the main method to keep the fluid atomized. For very viscous fluids, increase fluid pressure and nozzle diameter can help in this regard. It is noted from the above discussion that all the presented results so far are based on one inlet swirl velocity profile, namely Rt = 0.9. The inlet swirl velocity profile can also influence the swirl decay. These effects are shown in Fig. 10 for Case C. For each inlet swirl profile, swirl decay results along the pipe are plotted for three Reynolds numbers. The variation trends for the swirl decay are very similar for different inlet profiles. The Reynolds number plays an important role in the decay. However, there do exist differences in the decay behavior among different inlet swirl profiles. For the same Reynolds number, a higher value of Rt leads to a faster decay. This effect is more obvious in the plots of SðReÞ for different values of Rt at the Sð0Þ same distance from the inlet as shown in Fig. 11. This effect can be explained by the wall shear stress in the tangential direction. As seen in the governing equations, the decay of swirl motion in a straight pipe is mainly due to the wall drag. The wall drag depends on the swirl velocity gradient at the wall. Based on the definition of the inlet swirl velocity profiles, a higher
Fig. 11. The swirl number ratio
SðReÞ Sð0Þ
as a function of the Reynolds number of Case C at Z = 15 for Rt = 0.25, 0.50, 0.75 and 0.90.
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value of the transition radius (Rt) results in a higher velocity gradient at the wall in the tangential direction. Therefore, the swirl decays faster for a higher value of Rt. As a side note, the current analysis provides an analytical solution for swirl decay in a straight pipe, but it does have its limitations. This analysis can only be applied to steady laminar flows with developed axial velocity assuming no radial velocity. The entrance transition and unsteady effects are not considered. As the flow goes faster, turbulence may occur and the analysis will not be valid. 4. Conclusion In this paper, the laminar swirl flow in a straight pipe is revisited and analytical solutions are obtained by using a prescribed axial velocity profile. The simplified governing equations are solved by the separation of variable technique for a uniform slug flow profile and a parabolic velocity profile. The solutions are expressed by the Bessel function for a slug axial flow velocity and by the generalized Laguerre function for the parabolic velocity profile. The effects of the Reynolds numbers and the inlet swirl profiles on the swirl velocity distribution and swirl intensity decay along the pipe are analyzed and discussed. The swirl decay along the pipe is obtained in an explicit analytical form. Results suggest that a high Reynolds number leads to a slow decay of the swirl velocity and the swirl intensity and a high transition radius of the inlet swirl profile results in faster decay of the swirl due to the stronger wall drag effects. For a higher Reynolds number, the effect of the axial swirl diffusion term is negligible. For the two different axial velocity profiles, the swirl velocity and intensity decay quite differently. In addition, because for the entrance region of a laminar swirl flow, the axial velocity profile would transition from a slug flow to a developed flow, the current results can provide estimated bounds for the swirl velocity and the swirl decay in the pipe. Acknowledgement We greatly appreciate the financial support from the NC Space Grant and MeadWestvaco Corporation on projects investigating liquid atomization of swirl atomizers. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Bradley D. The hydrocyclone. New York: Pergamon Press Ltd.,; 1965. Lefebvre AH. Atomization and sprays. Taylor and Francis; 1989. Taylor GI. The boundary layer in the converging nozzle of a swirl atomizer. Quart J Mech Appl Math 1950;3:129–39. Binnie AM, Harris DP. The application of boundary layer theory to swirling liquid flow through a nozzle. Quart J Mech Appl Math 1950;3:89–106. Talbot L. Laminar swirling pipe flow. J Appl Mech 1954;21:1–7. Kreith F, Sonju OK. The decay of a turbulent swirl in a pipe. J Fluid Mech 1965;22(2):257–71. Kiya M, Shoichiro F, Mikio A. Laminar swirling flow in the entrance region of a circular pipe. Bull JSME 1971;14(73):659. Singh MP, Sinha PC, Aggarwal M. Swirling flow in a straight circular pipe. ZAMM 1980;60:429–36. Padmanabhan N. Swirling flow of a fluid in a straight tube. Indian J Pure Appl Mat 1984;15(4):943–50. Reader-Harris MJ. The decay of swirl in a pipe. Int J Heat Fluid Fl 1994;15(3):212–7. Yu SCM, Kitoh O. A general formulation for the decay of swirling motion along a straight pipe. Int Commun Heat Mass 1994;21(5):719–28. Najafi AF, Saidi MH, Sadeghipour MS, Souhar M. Boundary layer solution for the turbulent swirling decay flow through a fixed pipe: SBR at the inlet. Int J Eng Sci 2005;43:107–20. Maddahian R, Kebriaee A, Farhanieh B, Firoozabadi B. Analytical investigation of boundary layer growth and swirl intensity decay rate in a pipe. Arch Appl Mech, in press. doi:10.1007/s00419-010-0424-9. Ayinde TF. A generalized relationship for swirl decay in laminar pipe flow. Sadhana 2010;35(2):129–37. White FM. Viscous fluid flow. 2nd ed. New York: McGraw-Hill Company; 1991. Lucca-Negro O, O’Dohery T. Vortex breakdown: a review. Prog Energ Combust 2001;27:431–81. Alekseenko SV, Kuibin PA, Okulov VL, Shtork SI. Helical vortices in swirl flow. J Fluid Mech 1999;382:195–243. Wolfram S. Mathematica – A system for doing mathematics by computer. 2nd ed. New York: Addison-Wesley Publishing Company, Inc.; 1993. Kaplan W. Advanced mathematics for engineers. Addison-Wesley Publishing Company, Inc.; 1981. pp. 173–176.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Analytical solutions of laminar swirl decay in a straight pipe Shanshan Yao, Tiegang Fang ⇑ Department of Mechanical and Aerospace Engineering, North Carolina State University, 911 Oval Drive, Campus Box 7910, Raleigh, NC 27695, United States
a r t i c l e
i n f o
Article history: Received 4 March 2011 Received in revised form 22 November 2011 Accepted 30 November 2011 Available online 13 December 2011 Keywords: Laminar swirl flow Pipe flow Swirl nozzle Analytical solutions Developed flow
a b s t r a c t In this work, the laminar swirl flow in a straight pipe is revisited and solved analytically by using prescribed axial flow velocity profiles. Based on two axial velocity profiles, namely a slug flow and a developed parabolic velocity profiles, the swirl velocity equation is solved by the separation of variable technique for a rather general inlet swirl velocity distribution, which includes a forced vortex in the core and a free vortex near the wall. The solutions are expressed by the Bessel function for the slug flow and by the generalized Laguerre function for the developed parabolic velocity. Numerical examples are calculated and plotted for different combinations of influential parameters. The effects of the Reynolds number, the pipe axial distance, and the inlet swirl profiles on the swirl velocity distribution and the swirl decay are analyzed. The current results offer analytical equations to estimate the decay rate and the outlet swirl intensity and velocity distribution for the design of swirl flow devices. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Swirl flows have many important engineering applications such as the cyclone for separation of solid, liquid, and gas, swirl atomizers, swirl combustion devices, heat transfer enhancement, and others [1,2]. In most of these devices, swirl flow transport in a straight pipe is the most commonly used. Due to the wall effects inside the pipe, the swirl intensity decays as it moves downstream. Meanwhile, most of the applications require sufficiently large swirl intensity to maintain the performance of the devices and reduction of swirl decay becomes important. Therefore, the understanding and controlling the swirl transport in a pipe is of great importance in the design of these devices. The study of swirl decay in a pipe has a relatively long history. Some early works on the swirling flows related to a swirl atomizer were carried out with analytical techniques [3,4], where the boundary layer problem was solved for flow in a converging nozzle of a swirl atomizer. Talbot might be the first one who solved the laminar swirl flow in a pipe [5]. An early work on turbulent swirl decay in a pipe was performed by Kreith and Sonju [6], where an approximation solution was presented for a developed axial turbulent flow. The results were compared with experiments and good agreement was obtained in the region less than 20 diameters of the inlet. Later, theoretical analyses were conducted on laminar swirl flows in a pipe following the work of Talbot. Kiya et al. [7] studied the developing swirl flow in a pipe. The governing equations were solved numerically using finite difference method. A solid-body swirl was used in the inlet of the pipe. The flow development in the entrance region was solved. Singh et al. [8] analyzed the swirl flow development in the near inlet region based on boundary layer assumption and the problem was solved using similarity transformation for the swirl velocity near the wall. Series solutions for the same governing equations used in Ref. [8] were solved by series expansion and the wall shear stresses were calculated for blood swirl transport in a straight tube [9]. An analytical solution was presented by Reader-Harris [10] for a developed turbulent swirling pipe flow using the separation of variable technique and the solution was expressed by the Bessel functions. The swirl decay rate was given by a simple equation as a function of the distance along the pipe. An ⇑ Corresponding author. E-mail address: [email protected] (T. Fang). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.11.038
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analytical approach was used by Yu and Kitoh [11] to derive the swirl decay behavior for a straight pipe based on the conservation of angular momentum flux. Approximate solution was derived and good agreement with experiments was observed. One important finding is that the swirl decay in a straight pipe does not always following the exponential law and the decaying rate can depend on the downstream pipe wall drag coefficient, the flow Reynolds number, and the swirl intensity. Based on boundary layer assumption, the developing turbulent swirling flow through a fixed pipe was investigated numerically and analytically [12]. Recently, an integral analysis was performed for the entrance region based on boundary layer assumptions and the swirl decay rate was developed by Maddahian et al. [13]. Their results showed good agreement with the available experimental data. Ayinde [14] conducted numerical calculations on the laminar swirl flow in a pipe with a three-dimensional computational fluid dynamics (CFD) code and a swirl decay rate formula was proposed accounting the effects of the flow Reynolds number, the pipe length, inlet swirl intensity, and the inlet swirl velocity profile. The author showed the effects of swirl on the axial velocity distribution as the swirl transports along the pipe, however, the assumed inlet developed axial velocity can be recovered at the exit of the pipe. The velocity profiles between the inlet and the outlet falls between the slug flow profile and the fully developed parabolic velocity profiles. Based on the above discussion, although the laminar pipe flow has received significant attention, analytical solutions to this problem are not addressed sufficiently in the literature. In this work, the laminar swirl pipe flow is revisited analytically with exact solutions. In order to make the governing equations solvable exactly, it is assumed the axial velocity profile is developed and does not change along the pipe. The full swirl velocity equation will be solved for a slug axial flow and approximated swirl velocity equation will be solved for both slug flow and the developed parabolic velocity profiles. These solutions are presented explicitly and the swirl decay can be easily computed based on the results. In addition, because for the entrance region of a laminar swirl flow, the axial velocity profile would transition from a slug flow to a developed parabolic flow, the current results can provide estimated bounds for the swirl velocity and the swirl decay in the pipe. 2. Mathematical formulation and analytical solutions In this paper, we consider a laminar swirling flow in a straight pipe with a given swirling velocity profile at the inlet of the pipe. For incompressible fluids without body force and based on the axisymmetric flow assumption, the three-dimensional steady state Navier–Stokes (NS) equations in cylindrical coordinates read [15]
1 @ @uz ðrur Þ þ ¼ 0; r @r @z
ð1Þ
! @ur @ur u2h 1 @p @ 2 ur 1 @ur @ 2 ur ur þm þ þ uz ¼ þ 2 2 ; ur r @r @r @z r q @r @r 2 @z r ! 2 2 @uh @uh ur uh @ uh 1 @uh @ uh uh ur þ þ uz þ ¼m þ 2 2 ; r @r @r @z r @r 2 @z r ! 2 2 @uz @uz 1 @p @ uz 1 @uz @ uz ur þ þm þ uz ¼ þ 2 ; r @r @r @z q @z @r 2 @z
ð2aÞ ð2bÞ ð2cÞ
*
where the velocity vector is V ¼ ður ; uh ; uz Þ; m is the kinetic viscosity, p is the fluid pressure, and q is the density of the fluid. The three components in the cylindrical coordinates are r, h, and z. We assume the axial flow is developed and the axial flow velocity is specified as a function of r only, namely uz = uz(r). In addition, to solve the equation analytically using separation of 2 variables, we assume @@zu2h is small for a general axial flow uz = uz(r) at the beginning. But for a slug flow, we will consider all the terms in the swirl velocity later. Based on this assumption, it is obtained ur = 0 from the continuity equation and the governing equation for uh become
! @uh @ 2 uh 1 @uh uh uz ðrÞ þ ¼m 2 ; r @r @z @r 2 r
ð3Þ
with boundary conditions as
uh ðz; 0Þ ¼ 0;
uh ðz; aÞ ¼ 0;
uh ð0; rÞ ¼ uh;i ðrÞ;
ð4a; 4b; 4cÞ
where a is the pipe radius, and uh,i(r) is the inlet swirl velocity and in general, it is given as [14]
8 r < rt ; < uh;i;max rrt ; h i uh;i ðrÞ ¼ ar : uh;i;max rrt ar ; r P rt ; t
ð5Þ
where rt is the radial location for the swirl flow transitions from a forced vortex to a free vortex. The flow configuration is illustrated in Fig. 1. Before solving the swirling velocity equation, we define some dimensionless variables and velocities as
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Fig. 1. A schematic of the flow configuration.
z Z¼ ; a
r R¼ ; a
UðRÞ ¼
uz ; Ur
WðZ; RÞ ¼
uh ; Ur
ð6a; 6b; 6c; 6dÞ
where Ur is the reference velocity and it is the average axial velocity in the pipe. The governing equation (3) becomes
ReU
@W @ 2 W @ W ¼ ; þ @Z @R2 @R R
ð7Þ
with the boundary conditions as
WðZ; 0Þ ¼ 0;
WðZ; 1Þ ¼ 0;
Wð0; RÞ ¼
uh;i ðRÞ ; Ur
ð8Þ
where Re is the Reynolds number defined as Re ¼ aUm r . Since the swirling flow equation is linear, without of the loss of generality, we can set
8 < RRt ; R < Rt ; Wð0; RÞ ¼ R h 1R i : t ; R P Rt : R 1Rt
ð9Þ
To investigate the decay of the swirl intensity, a swirl number is defined as [13,16,17]
Ra S¼
uh uz r2 dr Ra ¼ SðzÞ: a 0 u2z rdr 0
ð10Þ
Substituting the dimensionless functions, the swirl number can be rewritten as
SðZÞ ¼
uh;i;max Ur
R1 0
WðZ; RÞUðRÞR2 dR : R1 UðRÞ2 RdR 0
ð11Þ
Then ratio of the swirl number at Z to the inlet swirl number is given by
R1 WðZ; RÞUðRÞR2 dR SðZÞ : ¼ R01 Sð0Þ Wð0; RÞUðRÞR2 dR
ð12Þ
0
d Based on Eq. (12), the swirl decay rate along the axial direction can be given as dZ
SðZÞ Sð0Þ
. Since the swirling velocity equation is
a linear partial differential equation, it can be solved by a separation of variable technique. Defining W(Z, R) = F(Z)G(R) and substituting it into Eq. (7) yield
_ ReUðRÞGðRÞFðzÞ ¼ FðZÞG00 ðRÞ þ FðZÞ
0 GðRÞ R
ð13Þ
Where a ‘‘dot’’ denotes a derivative with respect to Z, while a ‘‘prime’’ indicates a derivative with respect to R. By dividing both sides by F(Z)G(R) and separating variables, it is obtained
h i0 G00 ðRÞ þ GðRÞ _ R ReFðZÞ ¼ ¼ k2 FðZÞ UðRÞGðRÞ
ð14Þ
where k2 is a constant (called eigenvalue) to be determined later and k is a positive real number. The solution of F(Z) can be obtained as k2
FðZÞ ¼ e Re Z :
ð15Þ
The radial function G(R) becomes
R2 G00 þ RG0 þ ½k2 UðRÞR2 1G ¼ 0;
ð16Þ
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with boundary conditions
Gð0Þ ¼ 0 and Gð1Þ ¼ 0:
ð17a; 17bÞ
For a given axial velocity distribution, the solution of Eq. (16) can be obtained. In the current study, two typical axial flow profiles are used, namely the uniform slug flow and the developed parabolic velocity profile. Three cases are discussed as follows. 2.1 Case A: Slug flow with uniform velocity For a slug flow, U(R) = 1, Eq. (16) becomes
R2 G00 þ RG0 þ ½k2 R2 1G ¼ 0:
ð18Þ
The solution of Eq. (18) satisfying the boundary condition at R = 0 reads
GðRÞ ¼ Bessel Jð1; kRÞ:
ð19Þ
where Bessel J(a, x) is the Bessel function of the first kind [18]. The values of k, also called eigen-value, are determined by the boundary condition at the wall with R = 1,
Bessel Jð1; kn Þ ¼ 0:
ð20Þ
Based on the properties of the Bessel functions, there are an infinite number of eigen-values for this problem. The related solution of Eq. (19) for each given eigenvalue is called the eigen functions for this problem. Based on the Sturm–Liouville R1 theorem [19], the eigen-functions are orthogonal for any two given values of kn, namely 0 Bessel Jð1; km RÞ Bessel Jð1; kn RÞRdR ¼ 0 for m – n. The general solution of W(Z, R) reads
WðZ; RÞ ¼
1 X
k2 n
C n eRe Z Bessel Jð1; kn RÞ
ð21Þ
n¼1
Based on the orthogonality of the eigen-functions, the coefficient Cn is given by
R1 Cn ¼
0
Wð0; RÞBessel Jð1; kn RÞRdR : R1 Bessel Jð1; kn RÞ2 RdR 0
ð22Þ
2.2. Case B: Developed parabolic velocity profile For a developed parabolic velocity profile, uz(r) = 2Ur(1 R2). The governing equation for G(R) becomes
R2 G00 þ RG0 þ ½2k2 ð1 R2 ÞR2 1G ¼ 0;
ð23Þ
with the same boundary conditions as Eqs. (17a, 17b). The solution for Eq. (23) is
pffiffiffi Laguerre L 2pk ffiffi2 ; 1; 2kR2
pffi kR
2
e
2
GðRÞ ¼
R
ð24Þ
;
where Laguerre L(a, b, z) is called the generalized Laguerre function [18], and the boundary condition at R = 0 is satisfied automatically. By setting G(1) = 0, the eigen-values are given by
Laguerre L
pffiffiffi kn pffiffiffi ; 1; 2kn 2 2
¼ 0:
ð25Þ
The solution for the swirling flow is given by 2
WðZ; RÞ ¼
1 X n¼1
Cne
k2
Ren Z
knpRffi
e
2
Laguerre L
kpnffiffi ; 1; 2 2
R
pffiffiffi 2kn R2 :
ð26Þ
Based on the Sturm–Liouville theorem [19], the eigen-functions are orthogonal for any two given values of kn, namely km R2 k R2 pffiffi pffiffi pffi n pffi R 1 e 2 Laguerre L 2kpnffi2;1; 2kn R2 e 2 Laguerre L 2kpmffi2;1; 2km R2 Rð1 R2 ÞdR ¼ 0 for m – n. The coefficients in Eq. (26) can be obtained 0 R R based on the inlet swirling flow profile as
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R1 Cn ¼
0
k R2 p n 2 Laguerre L
ffi
e
ffi
kn p ;1; 2 2
pffiffi 2kn R2
Wð0; RÞ Rð1 R2 ÞdR R : 2 32 2 k R pffiffi pffi n kn 2 2 p ffi e Laguerre L ;1; 2 k R n R1 6 2 2 7 2 5 Rð1 R ÞdR 0 4 R
ð27Þ
2.3. Case C: Slug flow including axial swirl diffusion 2
In the previous derivation, we assume @@zu2h is small compared to the convection term. However, for the slug flow, we can include the axial diffusion term due to the simplicity of the velocity and the dimensionless swirl velocity equation becomes,
Re
@W @ 2 W @ W @2W þ : ¼ þ 2 @Z @R R @R @Z 2
ð28Þ
Following the previous procedure, we can separate the variables as
h i0 00 GðRÞ G ðRÞ þ _ € R ReFðZÞ FðzÞ ¼ ¼ k2n : FðZÞ GðRÞ
ð29Þ
We obtain the equation for the axial part,
F€ ReF_ k2n F ¼ 0:
ð30Þ
The solution reads
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
Re
FðZÞ ¼ e
4k þRe n 2
Z
ð31Þ
; 2
kn which can reduce to Eq. (12) for large Re, namely the slug flow solution without the axial diffusion term. However, Re is al pffiffiffiffiffiffiffiffiffiffiffiffiffi Re 4k2 þRe2 2 n for k > 1, which could lead to a faster swirl decay along the pipe for Case A than Case C as ways greater than n 2
shown in the result section. G(R) remains the same and all the eigen-functions and eigen-values keep the same as Case A. The general solution is given by
WðZ; RÞ ¼
1 X
Re
Cne
pffiffiffiffiffiffiffiffiffiffi ffi 2 2 4k þRe n 2
Z
Bessel Jð1; kn RÞ;
ð32Þ
n¼1
where the constant coefficients are given by Eq. (22). In order to obtain the swirling velocity distribution, the eigen-values for the two given axial velocity profiles must be solved first. All the presented results in this work were obtained using Mathematica 6.0 software. The eigen-values were calculated for both kinds of eigen-functions. In this work, n = 50 was used for all the calculations. Some examples of the eigenvalues, namely the first 30 eigen-values, are listed in Tables 1 and 2 for the Bessel function and the generalized Laguerre function, respectively. Before results are plotted and discussed, further analytical derivations are presented. Due to the simplicity of the Bessel R1 function, its integration can be expressed explicitly as 0 Bessel Jð1; kn RÞ2 RdR ¼ 12 Bessel Jð2; kn Þ2 . With the given inlet swirl flow profile (9), we can obtain
Z
( Rt Rt BesselJð0;Rt kn Þ BesselJð0;kn Þ BesselJð2;Rt kn Þ þ kn kn 1 Rt h i) 3 5 1 2 kn HypergeometricPFQ 2 ; 2; 2 ; 4 kn kn R3t HypergeometricPFQ 32 ; 2; 52 ; 14 k2n R2t ; 6
1
Wð0;RÞBesselJð1;kn RÞRdR ¼
0
Table 1 The first 30 eigen-values for the Bessel function of the first kind. n
kn
n
kn
n
kn
n
kn
n
kn
1 2 3 4 5 6
3.8317 7.0156 10.1735 13.3237 16.4706 19.6159
7 8 9 10 11 12
22.7601 25.9037 29.0468 32.1897 35.3323 38.4748
13 14 15 16 17 18
41.6171 44.7593 47.9015 51.0435 54.1856 57.3275
19 20 21 22 23 24
60.4695 63.6114 66.7532 69.8951 73.0369 76.1787
25 26 27 28 29 30
79.3205 82.4623 85.604 88.7458 91.8875 95.0292
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S. Yao, T. Fang / Commun Nonlinear Sci Numer Simulat 17 (2012) 3235–3246
Table 2 The first 30 eigen-values for the generalized Laguerre function. n
kn
n
kn
n
kn
n
kn
n
kn
1 2 3 4 5 6
3.2697 6.1147 8.9485 11.7794 14.6092 17.4386
7 8 9 10 11 12
20.2676 23.0965 25.9252 28.7539 31.5826 34.4111
13 14 15 16 17 18
37.2397 40.0683 42.8968 45.7253 48.5538 51.3823
19 20 21 22 23 24
54.2108 57.0392 59.8677 62.6962 65.5246 68.3531
25 26 27 28 29 30
71.1816 74.0100 76.8385 79.6669 82.4954 85.3238
where Hypergeometric PFQ[(a), (b, c), x] is the generalized hypergeometric function. Therefore, the constants can be given explicitly as Rt kn
Cn ¼
Rt Bessel Jð2; Rt kn Þ þ 1R t
Bessel Jð0;Rt kn ÞBessel Jð0;kn Þ kn
kn Hypergeometric PFQ ½ð32Þ;ð2;52Þ;14k2n kn R3t Hypergeometric PFQ ½ð32Þ;ð2;52Þ;14k2n R2t
6
:
1 Bessel Jð2; kn Þ2 2
ð33Þ
The above equation gives the coefficients for both Case A and Case C. Then the swirl number can be given for the slug flow as R1 WðZ;RÞR2 dR u 0 SðZÞ ¼ h;i;max . For Case A with the given inlet swirl velocity, we obtain 1 Ur 2
Z 1 n 1 1 k2 2uh;i;max X 2uh;i;max X C n eRe Z Bessel Jð2; kn Þ n SðZÞ ¼ C n e Re Z Bessel Jð1; kn RÞR2 dR ¼ : kn Ur Ur 0 n¼1 n¼1 k2
ð34Þ
For Case C, it is obtained Re
SðZÞ ¼
1 X Cn e
2uh;i;max Ur
pffiffiffiffiffiffiffiffiffiffi ffi 2 2 4k þRe n 2
n¼1
Z Bessel Jð2; kn Þ : kn
ð35Þ
The swirl number ratio is then given for Case A by
SðZÞ ¼ Sð0Þ
k2 nZ Re Bessel Jð2;kn Þ
P1
P1
Cn e
n¼1
P1
n¼1
kn C n Bessel Jð2;kn Þ kn
Cn e
k2 nZ Re Bessel Jð2;kn Þ
n¼1
¼
R1 0
kn
P1 ¼
Wð0; RÞR2 dR
Cn e
k2 nZ Re Bessel Jð2;kn Þ
n¼1
kn
Rt =6 þ R2t =6 R3t =12
ð36Þ
and for Case C by
SðZÞ ¼ Sð0Þ
P1
n¼1
Re
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
4k þRe n Z 2 Bessel Jð2;kn Þ
Cn e
R1 0
kn
pffiffiffiffiffiffiffiffiffiffi ffi 2 2
Cn e
4k þRe n Z 2 Bessel Jð2;kn Þ
n¼1
¼
Wð0; RÞR2 dR
Re
P1
kn
Rt =6 þ R2t =6 R3t =12
ð37Þ
:
For the parabolic velocity profiles, the swirl number becomes
SðZÞ ¼
uh;i;max Ur
P1
n¼1 C n e
k2 Ren Z
R1
k R2 p n 2 Laguerre L
ffi
e
0
ffi
kn p ;1; 2 2
Z 1e 1 k2 3 uh;i;max X n C n e Re Z 2 U r n¼1 0
2ð1 R2 ÞR2 dR
4ð1 R2 Þ2 RdR pffiffiffi ffi Laguerre L kpnffiffi ; 1; 2kn R2
2 knpR 2
¼
pffiffi 2kn R2
R
R1 0
2 2
R
2ð1 R2 ÞR2 dR:
ð38Þ
Due to the complexity of the generalized Laguerre function, the integration has to be evaluated numerically. The ratio of the swirl number reads
SðZÞ ¼ Sð0Þ
P1
n¼1 C n e
R1 0
e
k R2 p n 2 Laguerre L
R1 0
P1 ¼
k2
Ren Z
ffi
R 2
ffi
kn p ;1; 2 2
pffiffi 2kn R2
2ð1 R2 ÞR2 dR
2
Wð0; RÞR 2ð1 R ÞdR pffiffi ffi kn pffi ;1; 2kn R2 2 2
k R2 p n 2 Laguerre L e
R1 2ð1 R2 ÞR2 dR 0 R : 2 3 4 1 R 7 þ 7R 8R 3R þ 2R t t t t t 30
k2
Ren Z n¼1 C n e
ð39Þ
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Fig. 2. Comparison of the current analytical solution with the direct numerical simulation of the PDE for Case B under given conditions.
Fig. 3. Swirl velocity profiles of Case A for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
3. Results and discussion In this section, several numerical examples are presented and the effects of different parameters on the swirl velocity profiles and the swirl decay along the axial direction are analyzed. Before the current analytical solutions were applied,
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Fig. 4. Swirl velocity profiles of Case B for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
Fig. 5. Swirl velocity profiles of Case C for different values of pipe locations (top) and flow Reynolds numbers (bottom) for Rt = 0.9.
validation was carried out to show the correctness of these solutions. Comparison is made in Fig. 2 for given values of control parameters using the current analytical series solutions and direct numerical calculation (using Mathematica) of Eq. (7) for Case B. It is seen that the results are in good agreement. The swirl velocity profiles for the three cases are illustrated in Figs. 3–5 respectively for different axial locations and the flow Reynolds numbers. In each figure, the top plot shows the effects of
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the axial location from the inlet of the pipe on the swirl velocity distribution, while the effects of the Reynolds numbers are shown in the bottom plot. It is seen that for a given inlet swirl velocity distribution, namely Rt = 0.9 for the current examples, the magnitude of the swirl velocity becomes smaller with the increase in Z. Also the swirl velocity decay starts from the wall and penetrate into the center of the pipe with increasing the axial distance in the pipe. The effects of the flow Reynolds numbers on the swirl velocity, however, are quite different. The swirl velocity decays faster for a smaller Reynolds number when keeping the distance from the inlet the same. This is consistent with the physical observation. For the current problem, a higher value of Reynolds number implies a higher average axial velocity if the pipe diameter and the fluid are kept the same. This higher axial velocity transports the swirl motion downstream faster with less time for the wall to dampen the swirl velocity. A close scrutiny of the swirl velocity profiles reveals that there do exist some differences for the three cases. To further illustrate the differences among the three cases investigated in this paper, the swirl velocity profiles are compared for Cases A, B, and C with different combinations of Re and Z, shown in Fig. 6. There are four groups of parameters in the plots, including small Z and small Re, small Z and large Re, large Z and small Re, and large Z and large Re. All the plots are based on the same inlet swirl profile with Rt = 0.9. For a relatively small Re, the decay of the swirl velocity is very fast. The differences among the three cases are obvious. For Re = 10, the swirl velocities in Cases A and C decay faster than Case B in general. However, for a short distance from the inlet, Case B has a faster decay near the pipe wall in the near inlet region (Z = 1). This can be explained by the axial velocity distribution. For both Case A and Case C, the axial velocity near the wall is higher than that of Case B, say the parabolic velocity profile. This faster axial motion results in a slower decay of the swirl motion for a given distance from the inlet. In addition, for a given Re, Case A always decays faster than Case C as discussed before. The results for a high Re, however, are very different. With a higher Re, Case C and Case A give almost the same results and the curves are overlapped in the plot. Case B has a faster decay for the given values of Z due to a lower axial velocity in the near wall region. The swirl decay along the pipe, given by SðZÞ , is shown in Fig. 7 for Cases A and C for three values of the Reynolds numbers. Sð0Þ The results in Fig. 7 are consistent with the findings in Fig. 6. Case A leads to a faster swirl velocity decay compared with Case C. With the increase of the Reynolds numbers (Re > 100), the difference becomes very small. For Re < 10, the swirl velocity can be damped out within the distance of two pipe diameters. In order to reduce the decay of the swirl flow, a faster axial velocity and a large pipe or a less viscous fluid can help in this regard with increased flow Reynolds number. The decay difference between Case A and Case C can also be explained based on the governing equations and the flow configuration. For Case A, the axial swirl motion transport is only determined by the convective motion due to the axial flow velocity; while for Case C there is an extra axial diffusion contribution of swirl motion besides the convective swirl transport downstream along the pipe. A comparison of the swirl decay for Cases B and C is shown in Fig. 8. It is found that Case B provides a different swirl behavior. For a small value of Re, Case B decays faster than Case C, while for a relatively large Re, Case B decays slower than
Fig. 6. Comparison of the three cases for different combinations of the pipe location and the Reynolds number for Rt = 0.9.
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Fig. 7. The swirl decay as a function of the pipe locations for the three cases under different values of the Reynolds numbers for Rt = 0.9.
Fig. 8. The swirl decay as a function of the axial locations for Cases B and C under different values of the Reynolds numbers for Rt = 0.9.
Fig. 9. The swirl number ratio
SðReÞ Sð0Þ
as a function of the Reynolds number of Case C at different pipe locations for Rt = 0.9.
Case C (Re = 10 and Re = 100 in Fig. 8) for a large enough distance from the inlet. Near the inlet of the pipe, there exist a crossover between Cases B and C for a certain Re, which indicates that near the inlet Case B decays faster than Case C, as shown in the insert of Fig. 8. Based on the current comparisons among the three cases, it is concluded that the difference between Cases A and C becomes negligible for a sufficiently large Reynolds numbers. For small Reynolds numbers, both Cases A and B give overestimated swirl decay by neglecting the axial diffusion term, which can be comparable to the convective term. Therefore, for relatively large Re, Cases A and B provide good approximations for swirl velocity and its decay. In real physical applications, the velocity profiles in the pipe are not always developed and the interaction of the swirl with axial motion and radial motion of the fluids can affect the axial velocity profiles. However, the shape of axial velocity profiles is generally between the slug flow and the parabolic profile. Thus, Cases A and B can offer analytical upper and lower limits of swirl decay along the pipe.
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Fig. 10. The swirl decay as a function of the pipe locations for Case C under different values of the Reynolds numbers for Rt = 0.25, 0.50, 0.75 and 0.90.
To further illustrate the effects of the Reynolds numbers on the swirl decay, SðReÞ is plotted in Fig. 9 for different axial disSð0Þ tances from the inlet. It is confirmed that increasing the Reynolds number reduces the swirl decay and increasing distance from the inlet enhances the decay. This has important applications in the design of the swirl driven devices. For example, in the design of the swirl atomizer, in order to form a conical liquid sheet at the nozzle exit, sufficiently large swirl number should be maintained. For a given nozzle with designed diameter and length, increasing the flow Reynolds number is the main method to keep the fluid atomized. For very viscous fluids, increase fluid pressure and nozzle diameter can help in this regard. It is noted from the above discussion that all the presented results so far are based on one inlet swirl velocity profile, namely Rt = 0.9. The inlet swirl velocity profile can also influence the swirl decay. These effects are shown in Fig. 10 for Case C. For each inlet swirl profile, swirl decay results along the pipe are plotted for three Reynolds numbers. The variation trends for the swirl decay are very similar for different inlet profiles. The Reynolds number plays an important role in the decay. However, there do exist differences in the decay behavior among different inlet swirl profiles. For the same Reynolds number, a higher value of Rt leads to a faster decay. This effect is more obvious in the plots of SðReÞ for different values of Rt at the Sð0Þ same distance from the inlet as shown in Fig. 11. This effect can be explained by the wall shear stress in the tangential direction. As seen in the governing equations, the decay of swirl motion in a straight pipe is mainly due to the wall drag. The wall drag depends on the swirl velocity gradient at the wall. Based on the definition of the inlet swirl velocity profiles, a higher
Fig. 11. The swirl number ratio
SðReÞ Sð0Þ
as a function of the Reynolds number of Case C at Z = 15 for Rt = 0.25, 0.50, 0.75 and 0.90.
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value of the transition radius (Rt) results in a higher velocity gradient at the wall in the tangential direction. Therefore, the swirl decays faster for a higher value of Rt. As a side note, the current analysis provides an analytical solution for swirl decay in a straight pipe, but it does have its limitations. This analysis can only be applied to steady laminar flows with developed axial velocity assuming no radial velocity. The entrance transition and unsteady effects are not considered. As the flow goes faster, turbulence may occur and the analysis will not be valid. 4. Conclusion In this paper, the laminar swirl flow in a straight pipe is revisited and analytical solutions are obtained by using a prescribed axial velocity profile. The simplified governing equations are solved by the separation of variable technique for a uniform slug flow profile and a parabolic velocity profile. The solutions are expressed by the Bessel function for a slug axial flow velocity and by the generalized Laguerre function for the parabolic velocity profile. The effects of the Reynolds numbers and the inlet swirl profiles on the swirl velocity distribution and swirl intensity decay along the pipe are analyzed and discussed. The swirl decay along the pipe is obtained in an explicit analytical form. Results suggest that a high Reynolds number leads to a slow decay of the swirl velocity and the swirl intensity and a high transition radius of the inlet swirl profile results in faster decay of the swirl due to the stronger wall drag effects. For a higher Reynolds number, the effect of the axial swirl diffusion term is negligible. For the two different axial velocity profiles, the swirl velocity and intensity decay quite differently. In addition, because for the entrance region of a laminar swirl flow, the axial velocity profile would transition from a slug flow to a developed flow, the current results can provide estimated bounds for the swirl velocity and the swirl decay in the pipe. Acknowledgement We greatly appreciate the financial support from the NC Space Grant and MeadWestvaco Corporation on projects investigating liquid atomization of swirl atomizers. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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