A solution to the turbulent Graetz problem by matched asymptotic expansions for an axially rotating pipe subjected to external convection

International Journal of Heat and Mass Transfer 78 (2014) 901–907

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A solution to the turbulent Graetz problem by matched asymptotic expansions for an axially rotating pipe subjected to external convection B. Weigand a,⇑, R. Bogenfeld b a b

Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany Institut für Faserverbundleichtbau und Adaptronik, DLR Braunschweig Lilienthalplatz 7, 38108 Braunschweig, Germany

a r t i c l e

i n f o

Article history: Received 29 May 2014 Received in revised form 20 July 2014 Accepted 20 July 2014

Keywords: Convective heat transfer Graetz problem Turbulent flow Laminarization

a b s t r a c t Heat transfer in an axially rotating pipe subjected to turbulent internal flow is strongly suppressed by tube rotation. For increasing rotational speeds of the pipe wall, the flow starts to laminarize and eventually the axial velocity profile of the hydrodynamically fully developed flow approaches the one of a laminar pipe flow. The solution for the heat transfer of a fluid with a hydrodynamically fully developed velocity profile in an axially rotating pipe subjected to external convection leads to a Sturm–Liouville eigenvalue problem. Asymptotic expressions are shown for the eigenvalues and constants based on a matched asymptotic expansion approach. These values are compared with numerical calculations and good agreement is found. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Fluid flow and heat transfer in rotating pipes are not only of considerable theoretical interest, but also of great practical importance. Technical applications are for example a rotating power transmission shaft which is longitudinally bored and through which cooling air in a gas turbine is guided or a rotating heat exchanger. There exists a large number of publications in literature which are concerned with the flow and heat transfer in axially rotating pipes where only some few [1–5] are cited here. For a laminar inlet flow into an axially rotating pipe it was found that the rotation destabilizes the flow and the tangential velocity profile vu/vuw starts to deviate slightly from the linear distribution (a solid body rotation). In contrast, for a turbulent inlet flow into the pipe it has been observed in several publications [2,3] that the tangential velocity profile for the hydrodynamically fully developed turbulent flow can be assumed to be prescribed by vu/ vuw = (r/R)2. With this assumption the axial velocity profile can be predicted for the hydrodynamically fully developed flow by using a modified mixing length model with good agreement compared with experiments [2,3]. The heat transfer for thermal developing flow can then be predicted by analytically solving a Nusselt–Graetz problem. This has been done for the cases of constant wall temperature, constant wall heat flux and external convection [3–5] and good agreement with experimental data has been obtained. The problem with this kind of analytical solutions ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.07.057 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

of the Nusselt–Graetz problem is in general that the sums appearing in this sort of solutions might require a large number of eigenvalues and constants for the here considered cases, because of the fact that the flow undergoes a laminarization from the turbulent to the laminar flow state with increasing rotation rates N = Reu/ReD. This means also that the length of the thermal entrance region might increase to very large values for a flow with higher Reynolds numbers. The thermal entry length might then approach the one of a laminar flow, which is about Lth  0:05DReD Pr. Thus, asymptotic expressions for the eigenvalues and constants are very useful for these cases. Expressions for asymptotic eigenvalues have been developed for Nusselt–Graetz problems for stationary tubes in the past by several authors by using the method of matched asymptotic expansions. Sleicher et al. [6] and Notter and Sleicher [7] developed asymptotic expressions for turbulent pipe flow through a non-rotating pipe for the two boundary conditions of a constant wall temperature and a constant wall heat flux. In doing so, the area between pipe center and pipe wall had to be subdivided into three regions. Analytical solutions for the related eigenvalue problem have then been obtained for each of the regions and have then been matched. The velocity profile was not defined in their works by an analytic function, but the asymptotic expressions contain the gradient of the velocity distribution at the pipe wall. For a non-rotating tube with external convection asymptotic expressions have been obtained by Sadeghipour et al. [8]. However, in their expressions the velocity profile has been explicitly described for a non-rotating pipe with a fully developed turbulent velocity profile as a power law representation. Thus, these

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Nomenclature a cp D h ha N R k k NuD Pr Prt ReD Reu T T0 Tb Ta

thermal diffusivity = k/(qcp) [m2/s] specific heat at constant pressure [J/(kg K)] pipe diameter [m] heat transfer coefficient inside the pipe [W/(m2 K)] external heat transfer coefficient [W/(m2 K)] rotation rate = Reu/ReD [–] pipe radius [m] thermal conductivity [W/(m K)] modified eigenvalue [–] Nusselt number based on pipe diameter = hD/k [–] Prandtl number = m/a [–] turbulent Prandtl number = eh/em [–] Reynolds number based on D = v z D=m [–] rotational Reynolds number = vuwD/m [–] temperature [K] constant inlet temperature [K] bulk-temperature [K] constant ambient temperature [K]

expressions cannot be used for the current application where the flow field is strongly varying with rotation rate. The aim of the present paper is twofold. First we want to develop asymptotic expressions for the case of external convection for an axially rotating pipe. These expressions will be based on the excellent analysis of Sleicher et al. [6] and Notter and Sleicher [7]. The second aim of the paper is to show a rigorous comparison between eigenvalues, constants and Nusselt numbers predicted by matched asymptotic expansions and values from numerical solutions of the underlying Sturm–Liouville eigenvalue problem. Such comparisons have only been rarely shown before (see for example Weigand [9] for the case of constant wall temperature and constant wall heat flux) and show nicely the accuracy and usefulness of the matched asymptotic expansions for these complicated cases with external convection. Of course, such comparisons have been hardly possible at the time the asymptotic expressions in [6,7] have been developed. As a result of the analysis Nusselt number expressions are given here which can easily been calculated by using the first three numerically predicted eigenvalues and constants together with the approximations presented here.

vz; v u v z vuw z, r, u

axial and tangential velocity component [m/s] axial mean velocity [m/s] tangential velocity of the pipe wall [m/s] axial, radial and tangential coordinates [m]

Greek symbols eh eddy diffusivity [m2/s] em eddy kinematic viscosity [m2/s] q density [kg/m3] D external convection parameter = vR/k [–] kj eigenvalue [–] H dimensionless temperature [–] m kinematic viscosity [m2/s] # modified eigenfunction [–] v overall heat transfer coefficient [W/(m2 K)] Uj eigenfunction [–]

approach with increasing N the one of a laminar pipe flow [3]. Due to the constant fluid properties the equations of motion and the energy equation are decoupled. The temperature distribution and the heat transfer can be calculated by solving the energy equation for this case. Here we assume that the pipe is cooled externally by a constant ambient temperature Ta and a constant external heat transfer coefficient ha at the outside of the rotating pipe. The pipe has an outer radius Ra an inner radius R, the pipe wall has the constant thermal conductivity kw. This leads then to the overall heat transfer coefficient v, defined by

1

v

¼

  R R R þ ln h a Ra k w Ra

ð1Þ

The flow is assumed to be hydrodynamically fully developed and to enter the heated pipe section at z = 0 with a constant temperature T0. The energy equation is given in dimensionless form by

~rv~ z ð~r Þ

  @H @ @H ¼ Eð~rÞ~r @~r @ ~z @~r

ð2Þ

with the boundary conditions 2. Analysis Fig. 1 shows the geometrical configuration and the used coordinate system. The turbulent flow is assumed to be incompressible and hydrodynamically fully developed. The fluid properties are assumed to be constant. The axial velocity profiles for the hydrodynamically fully developed flow can be easily calculated (see [2,3]) and are shown in Fig. 2 together with some experimental results [5]. It is clearly visible how the turbulent flow gets laminarized with increasing rotation rate N and how the axial velocity profiles

~z ¼ 0 : H ¼ 1  T a =T 0 ~z > 0; ~r ¼ 0 : @ H=@~r ¼ 0 ~z > 0; ~r ¼ 1 : @ H=@~r þ DH ¼ 0

ð3Þ

and the dimensionless quantities

v uw T  Ta z 4 r vz ; ~z ¼ ; ~r ¼ ; v~ z ¼ ; N ¼ ; D ReD Pr R T0 v z v z v uw D hD v z D m vR Nu ¼ ; ReD ¼ ; Reu ¼ ; Pr ¼ ; D ¼ k k m m a

H¼

ð4Þ

The function Eð~r Þ contains the eddy diffusivity of the turbulent flow and has been modeled according to [2,3] by using a modified mixing length approach:

Pr Eð~r Þ ¼ 1 þ Prt

Fig. 1. Geometrical configuration and coordinate system (figure taken from [5]).

#1=2  2 " 2 @ v~ z 2 þ ð~r NÞ @~r

em ReD Pr l ¼1þ 2Prt R m

ð5Þ

The quantity D in the boundary condition given by Eq. (3) is a modified Biot number. For D ? 0 the case of an adiabatic wall is obtained, whereas for D ? 1 a constant wall temperature is achieved. Predictions with this method are in very good agreement

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Fig. 2. Axial velocity distribution as a function of the rotation rate N (figure taken from [5]).

with measured data [4]. Eq. (2) together with the boundary conditions (3) has been solved analytically by Weigand and Beer [5] by the method of separation of variables and eigenvalues and constants are given there. The solution for the Nusselt number is given by

Following Sleicher et al. [6] and Notter and Sleicher [7] the modified eigenfunction # can be prescribed by an asymptotic sequence given by

  P 2 ~ 2D 1 j¼1 Aj Uj ð1Þ exp kj z   P   NuD ¼ P Aj Uj ð1Þ  2 ~ 1  exp k2j ~z  1 1  2D 1 j¼1 k2 j¼1 Aj Uj ð1Þexp kj z

Introducing this expression into Eq. (9) and collecting terms of similar order in k, results in a sequence of linear differential equations. The procedure to develop the asymptotic solution for the eigenfunctions from these equations is relatively complicated, because the function w(n) in Eq. (9) has singularities in the centre of the pipe as well as at the wall. Thus the region between pipe center and pipe wall has to be subdivided into three areas (shown in Fig. 3). One region exists for each singularity and a middle region, providing a bridging between the regions with singularities. In the pipe center region and in the near wall region stretched local coordinates are introduced (center region: s = kn; near wall region: t = k(p  n)) as shown in Fig. 3. These coordinates are needed for afterwards matching the individual solutions. The middle region allows this matching by increasing the eigenvalue for fixed values of the coordinates. This leads to the later presented solution. For the three areas the function wðnÞ can be approximated according to [6] as follows

j

ð6Þ The Aj are constants and Uj ð~r Þ are eigenfunctions which are determined from the following eigenvalue problem [5]

  d dUj þ k2j v~ z ð~rÞ~r Uj ¼ 0 Eð~r Þ~r d~r d~r

ð7Þ

with the related homogeneous boundary conditions

dUj ¼0 d~r dUj þ DUj ¼ 0 d~r

~r ¼ 0 : ~r ¼ 1 :

ð8Þ

Because large numbers of eigenvalues (>200) are normally needed to get reliable results also for smaller axial distances a method of predicting the larger eigenvalues analytically is useful. Here the larger eigenvalues will be determined in the following with the method of matched asymptotic expansions. For this, the variables are changed in Eq. (7), so that the equation takes its ‘‘standard form’’ (see e.g. [9,10]) 2

d # dn

2

h i 2 þ k  wðnÞ # ¼ 0

1=2

#¼k

3=2

#0 þ k

3=2

ln k#1 þ k

Pipe center region : Middle region : Near wall region :

#2 þ   

wðnÞ ¼ 

1 4n2

;

ð11Þ

n ¼ p~r

wðnÞ ¼ 0

ð12Þ

5 1 b2 þ wðnÞ ¼  36 ðp  nÞ2 ðp  nÞ

ξ

ð9Þ

with the quantities

sffiffiffiffiffiffiffiffiffiffiffiffi Z sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi v~ z ~ 1 ~r v~ z  G¼ dr; k ¼ 2Gkj ; n ¼ dr G 0 p 0 2Eð~rÞ 2Eðr Þ  2   2    ~r v~ z Eð~r Þ 1=4 ~r v~ z Eð~rÞ 1=4 d2 ~r v~ z Eð~r Þ 1=4 Uj ; wðnÞ ¼ #¼ 2 2 2 dn2 1

Z

1

ð10Þ

Fig. 3. Areas in the pipe for the asymptotic analysis and the modified coordinates.

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where b2 is a constant for approximating the function w(n) by a Laurent series. A precise approximation of the singularities is a compelling requirement to achieving satisfying accuracy in the solution. In the middle region of the pipe the function w(n) has no singularities and can therefore be set to zero. For every region the solution of Eq. (9) can be obtained and the individual solutions have then to be matched. The solutions for the individual regions are quite lengthy and will therefore not been repeated here. The reader is referred to [6,7,10] for these solutions. The resulting solutions for the eigenfunctions are formally identical to the ones given by Sleicher et al. [6] and Notter and Sleicher [7] (which have been developed there for a constant wall temperature and a constant wall heat flux) except of the convective boundary condition at the pipe wall and the different turbulent eddy diffusivity for the axial rotating pipe. For more details on the solution method the reader is therefore referred to [6,7,10]. For the convective boundary conditions at the pipe wall one obtains after some calculation the following equation for the determination of the modified eigenvalue k

h   p p 1 C 1 2 sin kp   b2 k lnðkÞ cos kp  3  3  p 7 1  k cos kp  þ b2 lnðpÞ 3 18p     2p 2p 1 þ b2 k lnðkÞ cos kp  ¼ 2 sin kp  3 3    2p 7 1 þ b2 lnðpÞ þ k cos kp  18p 3

k ð3GÞ

2=3

2=3

H1=3 Cð2=3Þ

Cð4=3ÞðD  1=2Þ

;

H¼

ReD cf 1 @ v~ z ¼ 2 @~r W 64

ð13Þ

"pffiffiffi

# 1 3 þ kj ð1=2 þ C 1 Þ 187p þ b2 lnðkj pÞ 1 1 þ arctan p ffiffiffi

 1 3 p 1 þ 2C 1  kj 3=2 187p þ b2 lnðkj pÞ

D  12 1  þ 4=5 3 3þj D

@k

@ U0j

i

ð17Þ

~r¼1

The needed expressions in the denominator of Eq. (17) can be obtained from the analytical solution and are given by

   @ Uj 2G E0 @E0 3  4 ln k ¼  þ þ E 1 2 @k ~r¼1 ð3GHÞ1=6 Cð2=3Þk1=3 3k @k 3k  @E1 ln k 4E2 1 @E2  þ 2 k @k @k k 3k @ U0j @k

¼

~r ¼1

ð18Þ

G 1=3

1=6

ð3GHÞ Cð2=3Þk     E0 @E0 3  4 ln k @E1 ln k 4E2 1 @E2    þ þ E1 þ  2 2 k @k 3k @k @k k 3k 3k   1=6 1=3  2ðGHÞ k D0 @D0 3  4 ln k  5=6 þ þ D1 2 @k 3 Cð4=3Þ 3k 3k  @D1 ln k 4D2 1 @E2  ð19Þ  2þ k @k @k k 3k

 pffiffiffiffiffiffiffiffiffi p D0 ¼ 2 G=3 sin kp  ; E0 ¼ D0 3  pffiffiffiffiffiffiffiffiffi p D1 ¼ b2 G=3 cos kp  ; E1 ¼ D1 3   pffiffiffiffiffiffiffiffiffi 7 p D2 ¼  G=3 þ b2 ln p cos kp  ; 18p 3

ð20Þ E2 ¼ D2

3. Results and discussion

ð15Þ

with j being an integer j = 0, 1, 2, 3, . . . In addition the constant b2, which appears in Eqs. (12), (13) and (15) is a matching constant, which can be used, to improve the predictions from the matched asymptotic expansion by choosing this constant in such a way that for a low eigenvalue (e.g. 3th eigenvalue) the eigenvalue and the eigenconstants from the analytical prediction coincide with the numerical predictions. The reader is referred to the excellent paper of Notter and Sleicher [7], who introduced this concept, for a more detailed description. In order to obtain an explicit expression for the eigenvalues, the arctan-function in Eq. (15) can be approximated. For the case of a constant wall temperature or a constant wall heat flux this can be done by expanding the arctan-function and retaining only the first coefficient (see [6,7]). With the here used boundary condition of the third kind this is not possible anymore, because the convection parameter D can vary between zero and infinity. Thus, the arctan-function was approximated for various D and the following explicit approximate relation for the eigenvalues has been obtained

kj ¼ j þ

~r ¼1

þ

ð14Þ

The implicit Eq. (13) will provide an infinite number of eigenvalues. This equation can be rewritten in order to give an equation for the jth eigenvalues kj

kj ¼ j þ

pffiffiffi 2D

with the expressions

as an implicit equation. In this equation the following abbreviations have been used

C1 ¼

Bj ¼ h @U kj D @kj

ð16Þ

D

So we can use either the implicit (exact) Eq. (15) or the approximation (16) to obtain the asymptotic eigenvalues. The constants Bj = AjUj(1) needed for evaluating the Nusselt number given by Eq. (6) can be obtained according to Hsu [11] as

In order to obtain solutions of the energy Eq. (2), the turbulent Prandtl number appearing in Eq. (5) has to be specified. There is a large variety of different models in the literature prescribing the turbulent Prandtl number and the reader is referred to Kays et al. [12] or Weigand [10] for an overview of different models. Because one focus of the present paper is to compare results with the work of Sleicher et al. [6] and Notter and Sleicher [7] for a non-rotating pipe and for small Prandtl numbers, the model of Azer and Chao [14] has been used here. However, there is no restriction on the model for Prt and the above given analysis is not restricted to any model for the turbulent Prandtl number. The turbulent Prandtl number of Azer and Chao [14] is given by

Prt ¼

~0:25 Þ 1 þ Pe380 0:58 expðr D

~0:25 Þ 1 þ Re135 0:45 expðr

;

PeD ¼ ReD Pr

ð21Þ

D

Thus, all conclusions which will be drawn in the following will also hold when a different model for the turbulent Prandtl number is used instead of the above given one. 3.1. Numerical procedure and accuracy of the predictions The eigenvalues kj as well as the eigenfunctions Uj ð~rÞ were calculated numerically for the eigenvalue problem given by Eq. (7) by using a four-stage Runge–Kutta scheme. In order to examine the accuracy of the calculated values several calculations were carried out for laminar flows. The eigenvalues calculated here are in very good agreement with those of Shah and London [13] for laminar flow in a pipe. For turbulent flow, the eigenvalues and constants coincide with those reported by Weigand and Beer [5]. In addition the eigenvalues are in agreement with the predictions by Sleicher et al. [6] and Notter and Sleicher [7] for both limiting cases of a non-rotating pipe with either constant wall heat flux or a constant wall temperature. A comparison between the values of this work

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and the results obtained by Sleicher and Notter can be found in Table 1. This agreement is very good, as Eq. (15) for D ? 0 is converging to the eigenvalue equation for constant heat flux. To achieve additionally the agreement with the values published in their work like in Table 1, the appendant model for calculating the turbulent Prandtl number had to be applied. There’s also a good agreement between the results in this work and the analytical results obtained by Sadeghipour [8]. The corresponding Table 2 shows some values. Here it should be remarked, that the agreement is depending on the velocity profile of the hydrodynamically fully developed flow. In [8] an approximation for the velocity profile has been used. 3.2. Results First a comparison is given between numerically predicted eigenvalues and the asymptotic expressions obtained from Eqs. (15) and (16). These comparisons are provided for different modified Biot numbers D for a non-rotating pipe in Fig. 4 and Tables 3 and 4. It can be seen that the analytically predicted eigenvalues for the two Biot numbers D = 0.01 and D = 10 are nearly identical to the numerical calculations already for the third eigenvalue! It is also interesting to note that there is nearly no difference between the implicit (exact) Eq. (15) for the eigenvalues and the approximate Eq. (16). Of course the first and second eigenvalues show large differences between the numerical values and the ones obtained from the matched asymptotic expansion. This is clear, because in the matched asymptotic expansion we had to assume that the eigenvalues attain large values. In Fig. 5 a comparison of the eigenvalues is shown for a rotating pipe for D = 2 and N = 5. Again one can see the excellent agreement between analytical and numerical predictions. The simple explicit relation according to Eq. (16) gives nearly the same results as the implicit relation according to Eq. (15). In addition one can see that for increasing rotation rates even the first three eigenvalues start to be better predicted by the analytical method. This can be seen in Table 5, where eigenvalues are given for ReD = 50,000, D = 2 and N = 5. This is caused by the fact that the flow starts to

Table 1 Comparison between the eigenvalues calculated by Sleicher and Notter [7] for constant heat flux and the results in this work for D = 0.001. Pr

Re

0.0

10,000 50,000

0.01 0.06

Sleicher and

Present

Present

Notter k24

work k24num

work k24ana

344 345

351 353

342 343

10,000 500,000

368 1746

380 1751

367 1746

50,000 100,000

865 1480

894 1551

867 1472

Table 2 Comparison between the analytical eigenvalues of Sadeghipour [8] and this work. Parameters

Pr = 0.01 Re = 10,000 D = 0.1 Pr = 0.01 Re = 10,000 D=1

j

0 1 2 3 0 1 2 3

Sadeghipour: 0.194 14.608 48.117 100.553 1.525 16.457 50.097 102.61



2 k pjffiffi 2

Present

Present

work: k2j;num

work: k2j;ana:

0.194 15.162 49.152 102.514 1.504 16.470 50.110 102.57

1.000 17.296 53.262 103.614 1.678 17.614 52.098 105.25

Fig. 4. Eigenvalues for ReD = 10,000 for a non-rotating pipe (N = 0) and D = 0.01 and D = 25.

Table 3 Comparison between analytical ReD = 10,000, N = 0, D = 0.02.

and

numerical

predicted

eigenvalues

j

k2j;num

k2j;ana: Eq. (15)

k2j;ana: Eq. (16)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.02 15.40 50.69 105.79 180.68 275.34 389.77 523.93 677.84 851.47 1044.81 1257.87 1490.64 1743.12 2015.29 2307.17

0.99 17.16 53.10 108.75 184.11 279.15 393.88 528.29 682.39 856.17 1049.63 1262.77 1495.60 1748.10 2020.29 2312.16

1.00 17.16 53.10 108.75 184.11 279.15 393.88 528.29 682.39 856.17 1049.63 1262.77 1495.60 1748.10 2020.29 2312.16

for

laminarize and thus the singularities in the eigenvalue problem are less severe than in the case of N = 0. In the case for N = 2 the second eigenvalue deviates of the numerical predicted value by only 0.5% (for the implicit prediction) and by 0.9% (for the approximate equation). Even the first eigenvalue is predicted within 6% accuracy! The constants Bj = AjUj(1) which are needed for the prediction of the Nusselt number according to Eq. (6) are compared to the numerical predictions in Figs. 6 and 7 for different convective

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Table 4 Comparison between analytical ReD = 10,000, N = 0, D = 10.

and

numerical

predicted

eigenvalues

j

k2j;num

k2j;ana: Eq. (15)

k2j;ana: Eq. (16)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4.33 24.92 63.61 121.11 197.81 293.93 409.58 544.83 699.70 874.22 1068.40 1282.24 1515.75 1768.94 2041.80 2334.33

3.81 23.83 62.65 120.64 197.97 294.73 410.99 546.78 702.13 877.07 1071.61 1285.76 1519.53 1772.93 2045.96 2338.63

3.81 23.89 62.74 120.73 198.06 294.82 411.07 546.86 702.21 877.15 1071.68 1285.83 1519.60 1772.99 2046.02 2338.69

for

Fig. 6. Constants Bj: ReD = 10,000 and Pr = 0.02 for N = 2 and ReD = 20,000 for N = 5 with different D.

Fig. 5. Eigenvalues for ReD = 50,000 for a rotating pipe (N = 5) and D = 2.

Table 5 Comparison between analytical ReD = 50,000, N = 2, N = 5, D = 2.

and

numerical

predicted

eigenvalues

j

k2j;num

k2j;ana: Eq. (15)

k2j;ana: Eq. (16)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2.14 17.07 48.21 96.01 160.59 242.00 340.26 455.38 587.37 736.23 901.96 1084.57 1284.06 1500.42 1733.66 1983.78

2.15 17.20 48.04 95.91 159.93 241.60 339.41 454.71 586.87 735.92 901.63 1084.32 1283.83 1499.85 1733.21 1983.50

2.16 17.25 48.32 95.86 159.92 241.60 339.40 454.71 586.87 735.92 901.63 1084.32 1283.83 1499.85 1733.21 1983.50

for

parameters D and different rotation rates N. Fig. 6 depicts results for ReD = 10,000 with N = 2 and ReD = 20,000 with N = 5 for different D. Generally very good agreement can be seen. Only for very small values for D (nearly an adiabatic wall) small oscillations can be noted in the numerically predicted eigenconstants. These oscillations are based on the numerical accuracy for small j of

Fig. 7. Constants Bj: ReD = 50,000 and Pr = 0.02 for N = 2 and different D.

the eigenconstants. Hence a combined solution – numerically predicted kj , Bj for small values of j, analytically for large values of j – is more effective than both of these methods would be if applied separately. For predicting the constants Bj the quantity b2 which had been introduced in Eq. (12) had to be selected. This has been done in a way that the constants are identical to the numerical predicted values for j = 2. This improves the prediction quality as it has been shown by Notter and Sleicher [7] and Weigand [10]. Fig. 7 shows a comparison between the numerically predicted constants and

B. Weigand, R. Bogenfeld / International Journal of Heat and Mass Transfer 78 (2014) 901–907

907

4. Conclusions According to the present analytical study concerning the prediction of eigenvalues and eigenconstants for turbulent flow in a pipe with and without axial rotation, the following major conclusions can be drawn:  Eigenvalues can be predicted with excellent accuracy by using the matched asymptotic expansion method according to Sleicher et al. [6] and Notter and Sleicher [7] which has been extended in the present paper for a convective boundary condition at the pipe wall. This could be shown by comparing numerically predicted values with the analytical results for various D.  It has been shown that the method can easily be extended to rotating pipe flows and to boundary conditions of the third kind.  A numerical solution for small eigenvalues and the presented analytical procedure for larger values are complementing one another to an efficient combined solution, which is faster and more accurate than a purely numerical approach. In summary it can be concluded that the asymptotic predictions are able to give very reliable results for eigenvalues and constants even for the first eigenvalues. Conflict of interest None declared. References

Fig. 8. Comparison between numerically and analytically predicted local Nusselt numbers for ReD = 10,000 and Pr = 0.02 (N = 0 and N = 5) for various D.

the analytical predicted values for a larger Reynolds number ReD = 50,000, N = 2 and various values of D. Again the good agreement between analytically and numerically predicted values can be noted. Finally an exemplary prediction of the local Nusselt number is plotted in Fig. 8 for Re = 10,000, N = 0 and N = 5 for various values of D. By comparing the Nusselt numbers for N = 5 and N = 0 in Fig. 8 it can be noticed that the rotation results in flow laminarization and thus the Nusselt number decreases with increasing values of N. In addition, it is clearly visible that the analytical predicted local Nusselt numbers are in excellent agreement with the numerical values. Comparisons for the special cases of a constant wall temperature (D ? 1) and a constant wall heat flux can be found in Weigand [9] and show similar good agreement between numerical predictions and analytical results. The analytical predictions get slightly worse, if the Prandtl and/or the Reynolds number increase. However, as shown above, the predictions get closer to the numerical values when the rotation rate N increases. This is an important finding, because for higher rotation rates a larger number of eigenvalues and constants are needed, because of flow laminarization the thermal entrance region length increases.

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