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Transition phenomena and velocity distribution in constant-deceleration pipe flow
Experimental Thermal and Fluid Science 62 (2015) 175–182
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Transition phenomena and velocity distribution in constant-deceleration pipe flow Hisanori Saka a, Yoshiaki Ueda a, Kazuyoshi Nishihara b, Olusegun J. Ilegbusi c,⇑, Manabu Iguchi a a
Graduate School of Engineering, Osaka Electro-Communication University, 18-8 Hatsu-cho, Neyagawa, Osaka 572-8530, Japan Faculty of Engineering, Hokkaido University, North 13, West 8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan c Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, United States b
a r t i c l e
i n f o
Article history: Received 11 July 2014 Received in revised form 18 December 2014 Accepted 20 December 2014 Available online 30 December 2014 Keywords: Unsteady flow Pipe flow Constant-deceleration flow Turbulence transition
a b s t r a c t The critical Reynolds number for transition to turbulence in a constant-acceleration pipe flow is well known to be significantly higher than the value for steady pipe flow. This is a consequence of the significant suppression of amplification of disturbances entering the pipe by the acceleration in spite of the increase of inertial force due to increase in instantaneous Reynolds number. In contrast, relatively little is known of the transition phenomena in a constant-deceleration pipe flow. Such a flow system is investigated by an experimental method in which the cross-sectional mean velocity is initially kept constant and subsequently decreased linearly to zero. The characteristics of the transition phenomena will be determined by two factors – deceleration and viscosity. Deceleration will amplify the disturbances, while a decrease in the instantaneous Reynolds number is expected to suppress the amplification of the disturbances due to viscosity. The study uses hot-wire experiments to clarify which of the two effects predominates in constant-deceleration pipe flow. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Transition to turbulence in a constant acceleration pipe flow has been widely investigated [1–11]. In contrast, relatively limited information exists on transition to turbulence and reverse transition in a constant-deceleration pipe flow [2,3]. This paper considers experimental investigation of the latter type of flow, with emphasis on the relative contribution of deceleration and viscosity to the flow characteristics. Such a study will have significant practical relevance. For example, if transition to turbulence can be controlled by imposing unsteadiness such as periodic acceleration and deceleration on steady pipe flow, the energy required to deliver fluids may be reduced due to a decrease in the pipe frictional loss. The study could also enhance our understanding of fluid flow phenomena such as water hammer which results from rapid valve closure in pipeline systems. In the aforementioned studies on constant acceleration [1–11], the fluid in a pipe initially at rest is typically accelerated at a constant rate. The cross-sectional mean velocity, um, therefore is linearly increased from zero. The critical Reynolds number for the transition to turbulence, Recr, is equal to that in a steady pipe flow [12], Rest,cr, when the acceleration is lower than a certain
⇑ Corresponding author. http://dx.doi.org/10.1016/j.expthermflusci.2014.12.010 0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
critical value, while Recr becomes much higher than Rest,cr when the acceleration exceeds that critical value. The value for Rest,cr and consequently, the critical acceleration, is dependent on the extent of disturbances entering the pipe [11]. The critical Reynolds number, Recr often exceeds 1x106 [1–7], a condition attributed to the significant suppression of the disturbances under constant acceleration, although increased inertial force will likely amplify the disturbances. Previous studies have also found that there exist two types of transition patterns with respect to constant acceleration [1–4]. For example, flow transition has been found to occur simultaneously over the whole pipe due to flow instability when the constant acceleration is relatively large [3]. The other transition pattern appears under relatively small constant-acceleration condition. Flow transition occurs near the pipe inlet and forms a discontinuous surface propagating in the downstream direction with a velocity greater than the cross-sectional mean velocity. The limit between the two patterns is expressed by D/(2aT 2) = 1 103, where D is the pipe diameter, a is the constant acceleration, and T is the time required for complete opening of the valve. This critical value seems to be dependent on the disturbance level of the flow entering the pipe [11]. Further investigations are therefore still necessary to fully establish this critical value. In the present study of constant-deceleration phenomena, the cross-sectional mean velocity, um, is initially kept constant and
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Nomenclature a 0 a D J0 R Recr Rest Rest,cr
constant acceleration, m/s2 dimensionless constant acceleration = R3a/m2, – pipe diameter, m Bessel functions of the first kind, – pipe radius, m critical Reynolds number in unsteady pipe flow, – Reynolds number in steady pipe flow, – critical Reynolds number in steady pipe flow, -
then linearly decreased to zero, as illustrated in Fig. 1. Constant deceleration will amplify the disturbances entering the pipe, while viscous force will damp them. The objective of the study is to investigate which of the two effects predominates in a constantdeceleration pipe flow. The radial velocity distribution in a constant-deceleration turbulent flow has been found to deviate from the quasi-steady velocity distribution as the deceleration increases, and the frictional loss is greater than the quasi-steady value [2]. The critical Reynolds number for reverse transition to laminar flow (re-laminarization) is beyond the scope of the present study and not considered here. Three flow conditions are worth considering in such a study. First, when the initial steady flow is laminar, there are three possible scenarios- the constant-deceleration flow remains laminar, becomes transitional, or becomes turbulent. Second, when the initial steady flow is transitional, the following three conditions are possible- the constant deceleration flow reverts to laminar flow, remains transitional, or becomes turbulent. Finally, when the initial steady flow is turbulent, the possible conditions are similar to the latter i.e. the constant deceleration flow reverts to laminar flow, reverts to transitional flow, or remains turbulent. The last two initial flow conditions (transitional and turbulent) are considered in this study in order to understand the relative effects of constant deceleration and viscous force on the transition phenomena in a constant-deceleration pipe flow. 2. Experiment 2.1. Experimental apparatus and procedure
r p t u um usta x
m
radial distance measured from the pipe centerline, m pressure, Pa time, s axial velocity component, m/s cross-sectional mean velocity, m/s short-time averaged value of u, m/s axial distance measured from the pipe inlet, m kinematic viscosity, m2/s
of brass and had inner diameter, D, of 0.078 m and length, L, of 5.0 m. The origin of the cylindrical coordinates (x, r, h) in Fig. 2 is on the centerline at the inlet cross-section of the test pipe. A bell-mouth was connected upstream to direct air smoothly into the pipe. A device capable of generating unsteady flow of arbitrary waveform was also connected downstream of the test pipe [11]. The device consisted of a circular pipe of 0.05 m inner diameter and 0.8 m length equipped with a butterfly valve that was driven by a stepper motor. The valve could not rotate but could swing around its supporting rod at programmed speeds. The diameter of the tungsten hot-wire was 5 lm and its length was 2 mm. The frequency response of the hot-wire anemometer was 140 Kz for air flows. The anemometer was carefully calibrated near the inlet of the pipe using a precise Pitot tube system. As is widely known, the measurement of velocity data of less than about 0.5 m/s typically involves considerable uncertainty. However, such uncertainties have minimal effect on the present study due to our focus on the onset of turbulence. The measured velocity data were not filtered before digitization. The cross-sectional mean velocity, um, was obtained by averaging the time-averaged axial velocity component over the cross-section of the pipe. 2.2. Experiments performed The axial velocity component, u, is measured at axial crosssections ranging from x = 1.0 m (x/D = 12.8) to 4.0 m (x/D = 51.3) at equal intervals of 1.0 m, as shown in Fig. 2(a). There is provision for an I-probe that traverses in the vertical direction. The
The details of the experimental apparatus and procedure have been presented in a previous publication [11] and will only be described briefly here. Fig. 2 shows a schematic of the experimental apparatus using air as the working fluid. The test pipe was made
Fig. 1. Schematic of cross-sectional mean velocity history in a constant deceleration flow.
Fig. 2. Schematic of experimental apparatus.
177
cross-sectional mean velocity, um is measured at 20 radial positions (r/R = 0.00, 0.08, 0.15, 0.23, 0.31, 0.38, 0.46, 0.54, 0.62, 0.67, 0.72, 0.77, 0.82, 0.87, 0.90, 0.92, 0.95, 0.975, 0.986, and 0.990, where r is measured from the centerline towards the bottom wall of the pipe). The root-mean-square (RMS) value of the axial turbulence component, u’, can be calculated from the relation,
Dui ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xiþ25 ðuj usta;j Þ2 =ðN 1Þ j¼i25
usta;i ¼
iX þ25
ð1Þ
uj =N
ð2Þ
j¼i25
0
0
4
i N=51 Time t [s] (Arbitrary scale) Fig. 3. Determination of the short-time averaged axial velocity component.
12
Fig. 4. History of cross-sectional mean velocity used experiment.
constant cross-sectional mean velocity, denoted by um,st, and final stage of constant deceleration. The initial constant acceleration value is a = 1.75 m/s2. The flow is decelerated after it has reached steady state, with valve closure beginning at t = 12 s. Five steady-state conditions are considered, as shown in Fig. 5 - Rest = 13400, 11000, 8360, 5930, and 4190. The constant deceleration is chosen to be a = -2.8 m/s2. The time duration of the constant cross-sectional mean velocity used is approximately 10s in order to guarantee steady flow before the constantdeceleration stage. Fig. 1 shows that the history of the cross-sectional mean velocity, um, in the steady and constant deceleration stages can be expressed by,
um ¼ um;st ðt < t 1 Þ
ð3aÞ
¼ um:st þ at ðt1 6 t 6 t 2 Þ
ð3bÞ
where t is time and t1 = 12 s. Preliminary experiments were carried out to ensure repeatability of the measured data on the cross-sectional mean velocity, um. The um history was satisfactorily reproduced within a scatter of ±3%.
Run1(Re Case 1 st=13400)
um,st
0
8
Time t [s]
Cross−sectional mean velocity u m [m/s]
(Arbitrary scale)
where usta,i is the i -th short-time averaged value of the axial velocity component, u, and N (=51) is the number of A/D (Analog to digital) converted data on u (see Fig. 3). The term ‘‘short-time’’ implies a time scale that is much longer than the characteristic time of turbulence generation such as the burst period but much shorter than the closure time of the valve. The frequency of A/D conversion used is 2000 Hz. The details of the calculation method have been reported elsewhere [11]. The short time chosen for the study is (N-1)/2000 = 50/2000 = 0.025 s, which is much shorter than the valve closure time of about 0.5–1 s. The burst period of the transient flows considered in the study is unknown. Therefore a window size of 0.025 s is chosen so that it would be much smaller than the valve closure time, and can reproduce a designed velocity distribution of transient laminar flow. Accordingly, the window size thus determined can detect the onset of turbulence but the measured rms value may be dependent on the window size. Additional experiments are desirable to further understand the burst period in transient pipe flows and the effect of widow size on short-time rms values. The decision on whether the flow is laminar, transitional or turbulent flow is made based on the output signal of the axial velocity component, u, and the RMS value of the axial turbulence component, Du, calculated from Eq. (2). Fig. 4 shows a schematic illustration of the programmed history of the cross-sectional mean velocity, um, investigated. It consists of the initial stage of constant acceleration, subsequent stage of
Axial velocity u [m/s]
Cross−sectional mean velocity um [m/s] (Aribitrary scale)
H. Saka et al. / Experimental Thermal and Fluid Science 62 (2015) 175–182
Case 2 Run2(11000) Case 3 (8360) Run3 Case 4 (5930) Run4 Case 5 (4190) Run5
0 13
12
Time t [s] Fig. 5. Schematic of cross-sectional mean velocity.
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3. Experimental results and discussion
(a)
x = 4000mm, r ' = 0.00
3.1. Output signal of axial velocity component
3
R a
ð4Þ
m2
where m is the kinematic viscosity of fluid. This dimensionless acceleration parameter can be rewritten as, 0
a ¼
R3 a
m2
1 Da 1 ¼ Re2st 2 ¼ Re2st St 8 um;st 8
um;st D
Inertial force ¼ Rest ¼ Viscous force m Da a Temporal acceleration St ¼ 2 ¼ 2 ¼ Spatial acceleration um;st um;st =D
Axial velocity u [m/s]
11000 8360
2
5930 4190
0
2
4
6
8
10
12
14
16
Time t [s]
(b) x = 4000mm, r ' = 0.72 4
Re st 13400 11000 8360
2
5930 4190
ð5Þ
0 0
ð6Þ
2
4
6
8
10
12
14
16
Time t [s] ð7Þ
In the above equations, Rest is the Reynolds number in the initial steady state and St is the Strouhal number. A value of a0 = 7.38 105 was employed in the present study. As indicated previously, the viscous force was found to play a more significant role on transition than constant deceleration under the experimental conditions considered. 3.2. Radial distribution of axial velocity component at initial steady state 3.2.1. Laminar flow The radial distribution of the axial velocity component was measured to confirm the establishment of laminar steady flow before the deceleration stage. As mentioned above, the flow before the deceleration stage in Case 4 is transitional, i.e. laminar and turbulent phases alternate with irregular intervals. The entrance length, Le, in a steady laminar pipe flow is given by [13],
Le 0:05Rest D
Re st 13400
0
ð8Þ
(c)
Axial velocity u [m/s]
a0 ¼
4
Axial velocity u [m/s]
Fig. 6 shows the output signals of the axial velocity component, u, for the five Cases considered, measured at axial location x = 4.0 m (x/D = 51.3) and three dimensionless radial positions r/R = 0 (Fig. 6a), r/R = 0.72 (Fig. 6b), and r/R = 0.95 (Fig. 6c). The initial steady flow is turbulent in Cases 1 and 2 and transitional in the remaining Cases 3, 4, and 5. The latter set of cases is considered transitional because of the irregular appearance of intermittent turbulent slugs. Fig. 7 shows the axial velocity component measured at the three representative radial positions above at the steady-state and constant-deceleration stages. The corresponding RMS value of the axial turbulence component is shown in Fig. 8. The constant-deceleration flow remains turbulent in Cases 1 and 2 and transitional in Case 3. However, the flows in Cases 4 and 5 re-laminarize in the constant-deceleration stage as evidenced by the small (nearly zero) RMS values. A similar trend is observed at all other dimensionless radial positions in all the Cases. It can thus be concluded that the constant-deceleration flow is stabilized when the initial steady flow Reynolds number is in the range 4190–930. In other words, the effect of viscous force predominates over constant deceleration for the a 2.8 m/s2 considered. The following dimensionless acceleration, a’ has been introduced in previous studies to correlate the critical Reynolds number in a constant acceleration pipe flow [2,11]:
x = 4000mm, r ' = 0.95 4
Re st 13400
2
11000 8360 5930 4190
0 0
2
4
6
8
10
12
14
16
Time t [s] Fig. 6. Output signals of axial velocity component.
Substituting Rest = 5930 and D = 0.078 m into Eq. (8) yields
Le 0:05 4190 0:078 ¼ 23:1m
ð9Þ
Therefore, the axial range of measurement zone considered (x = 1.0–4.0 m) is within the developing flow region.
In order to confirm the establishment of laminar steady flow, the measured instantaneous velocity distributions in the laminar phase (Case 4) are compared with the approximate analytical
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(a) 4
(a)
x = 4000mm, r ' = 0.00
0.4
11000 2
8360 5930 4190
RMS value Δ u [m/s]
Axial velocity u [m/s]
Rest 13400
x = 4000mm, r ' = 0.00
Rest 0.2
13400 11000 8360 5930 4190
0
0 12
12
13
Time t [s]
(b)
(b)
x = 4000mm, r ' = 0.72
x = 4000mm, r ' = 0.72 0.4
Rest 13400 11000
2
8360 5930
RMS value Δ u [m/s]
4
Axial velocity u [m/s]
13
Time t [s]
Rest 0.2
13400 11000 8360 5930 4190
0
4190 0 12
12
13
Time t [s]
0.4
Re st 13400 11000 8360
RMS value Δ u [m/s]
Axial velocity u [m/s]
x = 4000mm, r ' = 0.95
2
13
(c)
(c) 4
Time t [s]
x = 4000mm, r ' = 0.95
Rest 0.2
13400 11000 8360 5930 4190
0
5930 4190
0
12
12
13
Time t [s]
13
Time t [s] Fig. 8. The RMS values of axial turbulence component near the deceleration stage.
Fig. 7. Output signals of axial velocity component near the deceleration stage.
solution proposed for steady developing region in a circular pipe [14]. The dimensionless axial velocity component, u0 is calculated from the following relations:
u0 ¼ / ð0 5 r 0 5 aÞ " 0 2 # r a 0 ða 5 r0 5 1Þ u ¼u 1 1a
ð10aÞ ð10bÞ
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where u0 and u are expressed by
u0 ¼
uR
ð11Þ
m
a u¼ 3 þ 2a þ a2
The radial distribution of usta is presented in Fig. 10. Also shown is the 1/nth power law distribution of Eq. (14). The results show that the measurement compares favorably with the 1/nth power law.
ð12Þ
The parameter a in the above equations is a given function of x/(DRest) [14]. The measured radial distribution of the dimensionless axial velocity component is shown in Fig. 9 as a function of axial locations. The results show that the axial velocity component in the laminar phase can be satisfactorily approximated by Eq. (10), implying that the flow is indeed steady under the conditions considered. 3.2.2. Turbulent flow in the initial steady state (Case 1) The entrance length, Le, in a steady turbulent pipe flow is expressed by [13],
Le 50D ¼ 50 0:078 ¼ 3:9m
ð13Þ
Thus, the measurement location x = 4.0 m for Case 1 is in the fully developed turbulent flow region. The short-time averaged axial velocity component at x = 4.0 m and t0 = 0.115 in Case 1 is compared with the following 1/nth power law velocity distribution proposed for steady turbulent pipe flow [13]: 1=n
u0sta ¼ u0sta;cl ð1 r 0 Þ
ð14Þ
where,
½ðn þ 1Þð2n þ 1Þ 0 u0sta;cl ¼ um 2n2 n ¼ 2 logðRest =10Þ
ð15Þ ð16Þ
3.3. Radial distribution of short-time averaged axial velocity component in the deceleration stage 3.3.1. Comparison of measured distribution with laminar analytical solution for fully developed flow in Case 4 The derivation of an analytical solution for laminar constantdeceleration flow in the developing region is complicated by the non-linearity of the governing equations. Therefore, as a first step, an analytical solution was derived for the fully developed region, although the axial measurement positions in Case 4 fall in the developing region. The solution was nevertheless compared with the measured velocity distribution in order to demonstrate that the two indeed differ. The equation of motion in the fully developed region can be expressed by [15],
@u0 @p0 @ 2 u0 1 @u0 þ þ 0 ¼ @t @x0 @r 02 r 0 @r0
ð17Þ
where the dimensionless quantities in Eq. (17) are defined as:
u0 ¼
Ru
m
;
x0 ¼
x ; R
t0 ¼
tm R
; 2
p0 ¼
pR2
qm2
;
r0 ¼
r R
ð18Þ
Here, p is the static pressure and q is the density of fluid. The solution of Eq. (17) is given by:
u0 ðr 0 ; t0 Þ ¼ u0m ð0ÞZ 1 ðr 0 ; t 0 Þ a01 Hðt 0 t 02 ÞZ 2 ðr 0 ; t 02 Þ þ a01 Hðt 0 t 01 ÞZ 2 ðr0 ; t 0 t 01 Þ
ð19Þ
Measured (Turbulent flow)
Case 4
Fig. 9. Dimensionless axial velocity component in steady laminar phase (Case 4).
Case 1
Fig. 10. Dimensionless axial velocity component in steady turbulent phase (Case 1).
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H. Saka et al. / Experimental Thermal and Fluid Science 62 (2015) 175–182
8000
where,
" Z m ðr 0 ; t 0 Þ :¼ W m ðr 0 ; t0 Þ þ
1 X
#
Gnm ðr 0 ; t0 Þ
6000
n¼1
ð21Þ ð22Þ ð23Þ
Eq. (19) is valid only for fully developed laminar flow. The measured velocity distributions in the developing region are compared with the analytical solution of Eq. (19) in Fig. 11. This figure demonstrates that this equation which was derived for fully-developed flow cannot predict the measured velocity distributions. The velocity distribution in the developing region is quite sensitive to the constant-deceleration condition and, as a result, a reverse flow occurs near the wall at the last stage of deceleration. The flow in the developing region is in general governed by both spatial acceleration and temporal acceleration. There is no evidence of reversed flow in the measured velocity distribution due to the dominant role of spatial acceleration over temporal acceleration. It should be remarked that the absence of reversed flow in the measurement cannot be associated with limitation of the experimental apparatus since a number of flow studies have confirmed that an I-probe can detect reverse flow although the output signals of the hot wire anemometer are positive [16–18].
4000
u'sta
W 1 ðr 0 ; t 0 Þ ¼ 2 1 r 02 1 1 r 02 1 3r 02 W 2 ðr 0 ; t 0 Þ ¼ 2 1 r 02 t0 24 4 exp y2n t 0 Gnm ðr 0 ; t 0 Þ ¼ ð1Þm ½J 0 ðyn Þ J 0 ðyn r 0 Þ y2m n J 0 ðyn Þ
t'=t'1+0.0000 t'=t'1+0.0015 t'=t'1+0.0035
ð20Þ
2000
0
–2000
0.8
0.6
0.4
0.2
0
r/R Fig. 12. Comparison of measured axial velocity component with numerical results in laminar deceleration flow (Case 4).
Measured (Turbulent flow)
3.3.2. Comparison of measured distribution with numerical simulation in Case 4 The transient laminar velocity distribution in the developing region was numerically simulated with the aid of FLUENT
Dimensionless axial velocity u’sta [-]
Case 1
Fig. 11. Comparison of measured axial velocity component with analytical solution in laminar deceleration flow (Case 4).
Fig. 13. Dimensionless axial velocity component in turbulent deceleration flow (Case 1).
commercial software. In the computational procedure, the pipe of diameter of D = 78 mm and length L = 10 m was discretized into 2.5 106 computational grids (32 32 grids in the cross-section). A segregated implicit solver and second-order upwind scheme were employed. A time-step of Dt = 2.5 103 s was used to achieve convergence at every iteration. The convergence of the computed solution was determined based on residuals set at 103 for the mass conservation and momentum equations. Fig. 12 shows the comparison of the laminar velocity distribution between the measured and computational results at axial location 4 m from the inlet. The results clearly show good agreement between the measurement and numerical prediction.
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Re 5930
α>α c 4190 α=−2.8m/s2
Time Fig. 14. Reverse transition to laminar flow due to application of repeated constantdeceleration and constant-acceleration (Constant-acceleration, a, must be greater than the critical value, ac, suppressing transition to turbulence.)
3.3.3. Comparison of measured velocity distribution with 1/nth power law for fully developed flow Fig. 13 presents the measured velocity distribution for Case 1 at three representative instants at the deceleration stage. Also shown are the corresponding analytical results for the 1/n-th power law. The figure shows that the 1/n-th power law satisfactorily predicts the measured velocity distributions for the conditions considered for Case 1. Although the scatter of the measured data seems to be rather high, such a scatter could be considered acceptable because velocity was measured only once at each radial position. A similar degree of agreement was also observed for Case 2 (not shown for brevity). A turbulent quasi-steady state condition can therefore be assumed to be established under the experimental conditions considered in Case 1. 3.3.4. Possible reverse transition to laminar flow Fig. 14 summarizes the axial velocity output signal obtained for a range of Reynolds number and different acceleration patterns. These results indicate that a transitional steady flow with Re in the range 4190–5930 can be re-laminarized by first applying a constant deceleration of a = 2.8 m/s2 followed by a constant acceleration greater than its critical value. Further experimental investigation is desirable and necessary to clarify if constant deceleration can similarly induce reverse transition from turbulent flow to laminar flow. 4. Conclusions Hot-wire experiments were performed in this study to clarify whether the effect of deceleration or viscosity, predominates in constant-deceleration pipe flow. Five experimental conditions were considered with steady state Reynolds numbers varying from 4190 to 13,400. The main findings of the study on the velocity distribution and transition to turbulence can be summarized as follows:
i. The effect of viscous force on flow transition is more dominant than that of constant deceleration when the Reynolds number in the initial steady flow is in the range 4190 to 5930. This implies that in this regime, a steady transitional flow becomes laminar at the constant-deceleration stage. ii. Laminar constant-deceleration flow is adequately predicted by numerical simulation. iii. Turbulent quasi-steady state occurs in the subsequent constant deceleration stage when the Reynolds number in the initial steady flow exceeds 11,000. iv. Viscous force suppresses the generation of turbulence against deceleration effect. This result, coupled with previous findings on transition to turbulence in constantacceleration pipe flow suggests that the frictional loss in pipe flows can be reduced by imposing unsteadiness. An example is the imposition of periodic constant acceleration and deceleration on steady pipe flows, as shown in Fig. 14.
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Transition phenomena and velocity distribution in constant-deceleration pipe flow Hisanori Saka a, Yoshiaki Ueda a, Kazuyoshi Nishihara b, Olusegun J. Ilegbusi c,⇑, Manabu Iguchi a a
Graduate School of Engineering, Osaka Electro-Communication University, 18-8 Hatsu-cho, Neyagawa, Osaka 572-8530, Japan Faculty of Engineering, Hokkaido University, North 13, West 8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan c Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, United States b
a r t i c l e
i n f o
Article history: Received 11 July 2014 Received in revised form 18 December 2014 Accepted 20 December 2014 Available online 30 December 2014 Keywords: Unsteady flow Pipe flow Constant-deceleration flow Turbulence transition
a b s t r a c t The critical Reynolds number for transition to turbulence in a constant-acceleration pipe flow is well known to be significantly higher than the value for steady pipe flow. This is a consequence of the significant suppression of amplification of disturbances entering the pipe by the acceleration in spite of the increase of inertial force due to increase in instantaneous Reynolds number. In contrast, relatively little is known of the transition phenomena in a constant-deceleration pipe flow. Such a flow system is investigated by an experimental method in which the cross-sectional mean velocity is initially kept constant and subsequently decreased linearly to zero. The characteristics of the transition phenomena will be determined by two factors – deceleration and viscosity. Deceleration will amplify the disturbances, while a decrease in the instantaneous Reynolds number is expected to suppress the amplification of the disturbances due to viscosity. The study uses hot-wire experiments to clarify which of the two effects predominates in constant-deceleration pipe flow. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Transition to turbulence in a constant acceleration pipe flow has been widely investigated [1–11]. In contrast, relatively limited information exists on transition to turbulence and reverse transition in a constant-deceleration pipe flow [2,3]. This paper considers experimental investigation of the latter type of flow, with emphasis on the relative contribution of deceleration and viscosity to the flow characteristics. Such a study will have significant practical relevance. For example, if transition to turbulence can be controlled by imposing unsteadiness such as periodic acceleration and deceleration on steady pipe flow, the energy required to deliver fluids may be reduced due to a decrease in the pipe frictional loss. The study could also enhance our understanding of fluid flow phenomena such as water hammer which results from rapid valve closure in pipeline systems. In the aforementioned studies on constant acceleration [1–11], the fluid in a pipe initially at rest is typically accelerated at a constant rate. The cross-sectional mean velocity, um, therefore is linearly increased from zero. The critical Reynolds number for the transition to turbulence, Recr, is equal to that in a steady pipe flow [12], Rest,cr, when the acceleration is lower than a certain
⇑ Corresponding author. http://dx.doi.org/10.1016/j.expthermflusci.2014.12.010 0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
critical value, while Recr becomes much higher than Rest,cr when the acceleration exceeds that critical value. The value for Rest,cr and consequently, the critical acceleration, is dependent on the extent of disturbances entering the pipe [11]. The critical Reynolds number, Recr often exceeds 1x106 [1–7], a condition attributed to the significant suppression of the disturbances under constant acceleration, although increased inertial force will likely amplify the disturbances. Previous studies have also found that there exist two types of transition patterns with respect to constant acceleration [1–4]. For example, flow transition has been found to occur simultaneously over the whole pipe due to flow instability when the constant acceleration is relatively large [3]. The other transition pattern appears under relatively small constant-acceleration condition. Flow transition occurs near the pipe inlet and forms a discontinuous surface propagating in the downstream direction with a velocity greater than the cross-sectional mean velocity. The limit between the two patterns is expressed by D/(2aT 2) = 1 103, where D is the pipe diameter, a is the constant acceleration, and T is the time required for complete opening of the valve. This critical value seems to be dependent on the disturbance level of the flow entering the pipe [11]. Further investigations are therefore still necessary to fully establish this critical value. In the present study of constant-deceleration phenomena, the cross-sectional mean velocity, um, is initially kept constant and
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Nomenclature a 0 a D J0 R Recr Rest Rest,cr
constant acceleration, m/s2 dimensionless constant acceleration = R3a/m2, – pipe diameter, m Bessel functions of the first kind, – pipe radius, m critical Reynolds number in unsteady pipe flow, – Reynolds number in steady pipe flow, – critical Reynolds number in steady pipe flow, -
then linearly decreased to zero, as illustrated in Fig. 1. Constant deceleration will amplify the disturbances entering the pipe, while viscous force will damp them. The objective of the study is to investigate which of the two effects predominates in a constantdeceleration pipe flow. The radial velocity distribution in a constant-deceleration turbulent flow has been found to deviate from the quasi-steady velocity distribution as the deceleration increases, and the frictional loss is greater than the quasi-steady value [2]. The critical Reynolds number for reverse transition to laminar flow (re-laminarization) is beyond the scope of the present study and not considered here. Three flow conditions are worth considering in such a study. First, when the initial steady flow is laminar, there are three possible scenarios- the constant-deceleration flow remains laminar, becomes transitional, or becomes turbulent. Second, when the initial steady flow is transitional, the following three conditions are possible- the constant deceleration flow reverts to laminar flow, remains transitional, or becomes turbulent. Finally, when the initial steady flow is turbulent, the possible conditions are similar to the latter i.e. the constant deceleration flow reverts to laminar flow, reverts to transitional flow, or remains turbulent. The last two initial flow conditions (transitional and turbulent) are considered in this study in order to understand the relative effects of constant deceleration and viscous force on the transition phenomena in a constant-deceleration pipe flow. 2. Experiment 2.1. Experimental apparatus and procedure
r p t u um usta x
m
radial distance measured from the pipe centerline, m pressure, Pa time, s axial velocity component, m/s cross-sectional mean velocity, m/s short-time averaged value of u, m/s axial distance measured from the pipe inlet, m kinematic viscosity, m2/s
of brass and had inner diameter, D, of 0.078 m and length, L, of 5.0 m. The origin of the cylindrical coordinates (x, r, h) in Fig. 2 is on the centerline at the inlet cross-section of the test pipe. A bell-mouth was connected upstream to direct air smoothly into the pipe. A device capable of generating unsteady flow of arbitrary waveform was also connected downstream of the test pipe [11]. The device consisted of a circular pipe of 0.05 m inner diameter and 0.8 m length equipped with a butterfly valve that was driven by a stepper motor. The valve could not rotate but could swing around its supporting rod at programmed speeds. The diameter of the tungsten hot-wire was 5 lm and its length was 2 mm. The frequency response of the hot-wire anemometer was 140 Kz for air flows. The anemometer was carefully calibrated near the inlet of the pipe using a precise Pitot tube system. As is widely known, the measurement of velocity data of less than about 0.5 m/s typically involves considerable uncertainty. However, such uncertainties have minimal effect on the present study due to our focus on the onset of turbulence. The measured velocity data were not filtered before digitization. The cross-sectional mean velocity, um, was obtained by averaging the time-averaged axial velocity component over the cross-section of the pipe. 2.2. Experiments performed The axial velocity component, u, is measured at axial crosssections ranging from x = 1.0 m (x/D = 12.8) to 4.0 m (x/D = 51.3) at equal intervals of 1.0 m, as shown in Fig. 2(a). There is provision for an I-probe that traverses in the vertical direction. The
The details of the experimental apparatus and procedure have been presented in a previous publication [11] and will only be described briefly here. Fig. 2 shows a schematic of the experimental apparatus using air as the working fluid. The test pipe was made
Fig. 1. Schematic of cross-sectional mean velocity history in a constant deceleration flow.
Fig. 2. Schematic of experimental apparatus.
177
cross-sectional mean velocity, um is measured at 20 radial positions (r/R = 0.00, 0.08, 0.15, 0.23, 0.31, 0.38, 0.46, 0.54, 0.62, 0.67, 0.72, 0.77, 0.82, 0.87, 0.90, 0.92, 0.95, 0.975, 0.986, and 0.990, where r is measured from the centerline towards the bottom wall of the pipe). The root-mean-square (RMS) value of the axial turbulence component, u’, can be calculated from the relation,
Dui ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xiþ25 ðuj usta;j Þ2 =ðN 1Þ j¼i25
usta;i ¼
iX þ25
ð1Þ
uj =N
ð2Þ
j¼i25
0
0
4
i N=51 Time t [s] (Arbitrary scale) Fig. 3. Determination of the short-time averaged axial velocity component.
12
Fig. 4. History of cross-sectional mean velocity used experiment.
constant cross-sectional mean velocity, denoted by um,st, and final stage of constant deceleration. The initial constant acceleration value is a = 1.75 m/s2. The flow is decelerated after it has reached steady state, with valve closure beginning at t = 12 s. Five steady-state conditions are considered, as shown in Fig. 5 - Rest = 13400, 11000, 8360, 5930, and 4190. The constant deceleration is chosen to be a = -2.8 m/s2. The time duration of the constant cross-sectional mean velocity used is approximately 10s in order to guarantee steady flow before the constantdeceleration stage. Fig. 1 shows that the history of the cross-sectional mean velocity, um, in the steady and constant deceleration stages can be expressed by,
um ¼ um;st ðt < t 1 Þ
ð3aÞ
¼ um:st þ at ðt1 6 t 6 t 2 Þ
ð3bÞ
where t is time and t1 = 12 s. Preliminary experiments were carried out to ensure repeatability of the measured data on the cross-sectional mean velocity, um. The um history was satisfactorily reproduced within a scatter of ±3%.
Run1(Re Case 1 st=13400)
um,st
0
8
Time t [s]
Cross−sectional mean velocity u m [m/s]
(Arbitrary scale)
where usta,i is the i -th short-time averaged value of the axial velocity component, u, and N (=51) is the number of A/D (Analog to digital) converted data on u (see Fig. 3). The term ‘‘short-time’’ implies a time scale that is much longer than the characteristic time of turbulence generation such as the burst period but much shorter than the closure time of the valve. The frequency of A/D conversion used is 2000 Hz. The details of the calculation method have been reported elsewhere [11]. The short time chosen for the study is (N-1)/2000 = 50/2000 = 0.025 s, which is much shorter than the valve closure time of about 0.5–1 s. The burst period of the transient flows considered in the study is unknown. Therefore a window size of 0.025 s is chosen so that it would be much smaller than the valve closure time, and can reproduce a designed velocity distribution of transient laminar flow. Accordingly, the window size thus determined can detect the onset of turbulence but the measured rms value may be dependent on the window size. Additional experiments are desirable to further understand the burst period in transient pipe flows and the effect of widow size on short-time rms values. The decision on whether the flow is laminar, transitional or turbulent flow is made based on the output signal of the axial velocity component, u, and the RMS value of the axial turbulence component, Du, calculated from Eq. (2). Fig. 4 shows a schematic illustration of the programmed history of the cross-sectional mean velocity, um, investigated. It consists of the initial stage of constant acceleration, subsequent stage of
Axial velocity u [m/s]
Cross−sectional mean velocity um [m/s] (Aribitrary scale)
H. Saka et al. / Experimental Thermal and Fluid Science 62 (2015) 175–182
Case 2 Run2(11000) Case 3 (8360) Run3 Case 4 (5930) Run4 Case 5 (4190) Run5
0 13
12
Time t [s] Fig. 5. Schematic of cross-sectional mean velocity.
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3. Experimental results and discussion
(a)
x = 4000mm, r ' = 0.00
3.1. Output signal of axial velocity component
3
R a
ð4Þ
m2
where m is the kinematic viscosity of fluid. This dimensionless acceleration parameter can be rewritten as, 0
a ¼
R3 a
m2
1 Da 1 ¼ Re2st 2 ¼ Re2st St 8 um;st 8
um;st D
Inertial force ¼ Rest ¼ Viscous force m Da a Temporal acceleration St ¼ 2 ¼ 2 ¼ Spatial acceleration um;st um;st =D
Axial velocity u [m/s]
11000 8360
2
5930 4190
0
2
4
6
8
10
12
14
16
Time t [s]
(b) x = 4000mm, r ' = 0.72 4
Re st 13400 11000 8360
2
5930 4190
ð5Þ
0 0
ð6Þ
2
4
6
8
10
12
14
16
Time t [s] ð7Þ
In the above equations, Rest is the Reynolds number in the initial steady state and St is the Strouhal number. A value of a0 = 7.38 105 was employed in the present study. As indicated previously, the viscous force was found to play a more significant role on transition than constant deceleration under the experimental conditions considered. 3.2. Radial distribution of axial velocity component at initial steady state 3.2.1. Laminar flow The radial distribution of the axial velocity component was measured to confirm the establishment of laminar steady flow before the deceleration stage. As mentioned above, the flow before the deceleration stage in Case 4 is transitional, i.e. laminar and turbulent phases alternate with irregular intervals. The entrance length, Le, in a steady laminar pipe flow is given by [13],
Le 0:05Rest D
Re st 13400
0
ð8Þ
(c)
Axial velocity u [m/s]
a0 ¼
4
Axial velocity u [m/s]
Fig. 6 shows the output signals of the axial velocity component, u, for the five Cases considered, measured at axial location x = 4.0 m (x/D = 51.3) and three dimensionless radial positions r/R = 0 (Fig. 6a), r/R = 0.72 (Fig. 6b), and r/R = 0.95 (Fig. 6c). The initial steady flow is turbulent in Cases 1 and 2 and transitional in the remaining Cases 3, 4, and 5. The latter set of cases is considered transitional because of the irregular appearance of intermittent turbulent slugs. Fig. 7 shows the axial velocity component measured at the three representative radial positions above at the steady-state and constant-deceleration stages. The corresponding RMS value of the axial turbulence component is shown in Fig. 8. The constant-deceleration flow remains turbulent in Cases 1 and 2 and transitional in Case 3. However, the flows in Cases 4 and 5 re-laminarize in the constant-deceleration stage as evidenced by the small (nearly zero) RMS values. A similar trend is observed at all other dimensionless radial positions in all the Cases. It can thus be concluded that the constant-deceleration flow is stabilized when the initial steady flow Reynolds number is in the range 4190–930. In other words, the effect of viscous force predominates over constant deceleration for the a 2.8 m/s2 considered. The following dimensionless acceleration, a’ has been introduced in previous studies to correlate the critical Reynolds number in a constant acceleration pipe flow [2,11]:
x = 4000mm, r ' = 0.95 4
Re st 13400
2
11000 8360 5930 4190
0 0
2
4
6
8
10
12
14
16
Time t [s] Fig. 6. Output signals of axial velocity component.
Substituting Rest = 5930 and D = 0.078 m into Eq. (8) yields
Le 0:05 4190 0:078 ¼ 23:1m
ð9Þ
Therefore, the axial range of measurement zone considered (x = 1.0–4.0 m) is within the developing flow region.
In order to confirm the establishment of laminar steady flow, the measured instantaneous velocity distributions in the laminar phase (Case 4) are compared with the approximate analytical
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(a) 4
(a)
x = 4000mm, r ' = 0.00
0.4
11000 2
8360 5930 4190
RMS value Δ u [m/s]
Axial velocity u [m/s]
Rest 13400
x = 4000mm, r ' = 0.00
Rest 0.2
13400 11000 8360 5930 4190
0
0 12
12
13
Time t [s]
(b)
(b)
x = 4000mm, r ' = 0.72
x = 4000mm, r ' = 0.72 0.4
Rest 13400 11000
2
8360 5930
RMS value Δ u [m/s]
4
Axial velocity u [m/s]
13
Time t [s]
Rest 0.2
13400 11000 8360 5930 4190
0
4190 0 12
12
13
Time t [s]
0.4
Re st 13400 11000 8360
RMS value Δ u [m/s]
Axial velocity u [m/s]
x = 4000mm, r ' = 0.95
2
13
(c)
(c) 4
Time t [s]
x = 4000mm, r ' = 0.95
Rest 0.2
13400 11000 8360 5930 4190
0
5930 4190
0
12
12
13
Time t [s]
13
Time t [s] Fig. 8. The RMS values of axial turbulence component near the deceleration stage.
Fig. 7. Output signals of axial velocity component near the deceleration stage.
solution proposed for steady developing region in a circular pipe [14]. The dimensionless axial velocity component, u0 is calculated from the following relations:
u0 ¼ / ð0 5 r 0 5 aÞ " 0 2 # r a 0 ða 5 r0 5 1Þ u ¼u 1 1a
ð10aÞ ð10bÞ
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where u0 and u are expressed by
u0 ¼
uR
ð11Þ
m
a u¼ 3 þ 2a þ a2
The radial distribution of usta is presented in Fig. 10. Also shown is the 1/nth power law distribution of Eq. (14). The results show that the measurement compares favorably with the 1/nth power law.
ð12Þ
The parameter a in the above equations is a given function of x/(DRest) [14]. The measured radial distribution of the dimensionless axial velocity component is shown in Fig. 9 as a function of axial locations. The results show that the axial velocity component in the laminar phase can be satisfactorily approximated by Eq. (10), implying that the flow is indeed steady under the conditions considered. 3.2.2. Turbulent flow in the initial steady state (Case 1) The entrance length, Le, in a steady turbulent pipe flow is expressed by [13],
Le 50D ¼ 50 0:078 ¼ 3:9m
ð13Þ
Thus, the measurement location x = 4.0 m for Case 1 is in the fully developed turbulent flow region. The short-time averaged axial velocity component at x = 4.0 m and t0 = 0.115 in Case 1 is compared with the following 1/nth power law velocity distribution proposed for steady turbulent pipe flow [13]: 1=n
u0sta ¼ u0sta;cl ð1 r 0 Þ
ð14Þ
where,
½ðn þ 1Þð2n þ 1Þ 0 u0sta;cl ¼ um 2n2 n ¼ 2 logðRest =10Þ
ð15Þ ð16Þ
3.3. Radial distribution of short-time averaged axial velocity component in the deceleration stage 3.3.1. Comparison of measured distribution with laminar analytical solution for fully developed flow in Case 4 The derivation of an analytical solution for laminar constantdeceleration flow in the developing region is complicated by the non-linearity of the governing equations. Therefore, as a first step, an analytical solution was derived for the fully developed region, although the axial measurement positions in Case 4 fall in the developing region. The solution was nevertheless compared with the measured velocity distribution in order to demonstrate that the two indeed differ. The equation of motion in the fully developed region can be expressed by [15],
@u0 @p0 @ 2 u0 1 @u0 þ þ 0 ¼ @t @x0 @r 02 r 0 @r0
ð17Þ
where the dimensionless quantities in Eq. (17) are defined as:
u0 ¼
Ru
m
;
x0 ¼
x ; R
t0 ¼
tm R
; 2
p0 ¼
pR2
qm2
;
r0 ¼
r R
ð18Þ
Here, p is the static pressure and q is the density of fluid. The solution of Eq. (17) is given by:
u0 ðr 0 ; t0 Þ ¼ u0m ð0ÞZ 1 ðr 0 ; t 0 Þ a01 Hðt 0 t 02 ÞZ 2 ðr 0 ; t 02 Þ þ a01 Hðt 0 t 01 ÞZ 2 ðr0 ; t 0 t 01 Þ
ð19Þ
Measured (Turbulent flow)
Case 4
Fig. 9. Dimensionless axial velocity component in steady laminar phase (Case 4).
Case 1
Fig. 10. Dimensionless axial velocity component in steady turbulent phase (Case 1).
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8000
where,
" Z m ðr 0 ; t 0 Þ :¼ W m ðr 0 ; t0 Þ þ
1 X
#
Gnm ðr 0 ; t0 Þ
6000
n¼1
ð21Þ ð22Þ ð23Þ
Eq. (19) is valid only for fully developed laminar flow. The measured velocity distributions in the developing region are compared with the analytical solution of Eq. (19) in Fig. 11. This figure demonstrates that this equation which was derived for fully-developed flow cannot predict the measured velocity distributions. The velocity distribution in the developing region is quite sensitive to the constant-deceleration condition and, as a result, a reverse flow occurs near the wall at the last stage of deceleration. The flow in the developing region is in general governed by both spatial acceleration and temporal acceleration. There is no evidence of reversed flow in the measured velocity distribution due to the dominant role of spatial acceleration over temporal acceleration. It should be remarked that the absence of reversed flow in the measurement cannot be associated with limitation of the experimental apparatus since a number of flow studies have confirmed that an I-probe can detect reverse flow although the output signals of the hot wire anemometer are positive [16–18].
4000
u'sta
W 1 ðr 0 ; t 0 Þ ¼ 2 1 r 02 1 1 r 02 1 3r 02 W 2 ðr 0 ; t 0 Þ ¼ 2 1 r 02 t0 24 4 exp y2n t 0 Gnm ðr 0 ; t 0 Þ ¼ ð1Þm ½J 0 ðyn Þ J 0 ðyn r 0 Þ y2m n J 0 ðyn Þ
t'=t'1+0.0000 t'=t'1+0.0015 t'=t'1+0.0035
ð20Þ
2000
0
–2000
0.8
0.6
0.4
0.2
0
r/R Fig. 12. Comparison of measured axial velocity component with numerical results in laminar deceleration flow (Case 4).
Measured (Turbulent flow)
3.3.2. Comparison of measured distribution with numerical simulation in Case 4 The transient laminar velocity distribution in the developing region was numerically simulated with the aid of FLUENT
Dimensionless axial velocity u’sta [-]
Case 1
Fig. 11. Comparison of measured axial velocity component with analytical solution in laminar deceleration flow (Case 4).
Fig. 13. Dimensionless axial velocity component in turbulent deceleration flow (Case 1).
commercial software. In the computational procedure, the pipe of diameter of D = 78 mm and length L = 10 m was discretized into 2.5 106 computational grids (32 32 grids in the cross-section). A segregated implicit solver and second-order upwind scheme were employed. A time-step of Dt = 2.5 103 s was used to achieve convergence at every iteration. The convergence of the computed solution was determined based on residuals set at 103 for the mass conservation and momentum equations. Fig. 12 shows the comparison of the laminar velocity distribution between the measured and computational results at axial location 4 m from the inlet. The results clearly show good agreement between the measurement and numerical prediction.
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Re 5930
α>α c 4190 α=−2.8m/s2
Time Fig. 14. Reverse transition to laminar flow due to application of repeated constantdeceleration and constant-acceleration (Constant-acceleration, a, must be greater than the critical value, ac, suppressing transition to turbulence.)
3.3.3. Comparison of measured velocity distribution with 1/nth power law for fully developed flow Fig. 13 presents the measured velocity distribution for Case 1 at three representative instants at the deceleration stage. Also shown are the corresponding analytical results for the 1/n-th power law. The figure shows that the 1/n-th power law satisfactorily predicts the measured velocity distributions for the conditions considered for Case 1. Although the scatter of the measured data seems to be rather high, such a scatter could be considered acceptable because velocity was measured only once at each radial position. A similar degree of agreement was also observed for Case 2 (not shown for brevity). A turbulent quasi-steady state condition can therefore be assumed to be established under the experimental conditions considered in Case 1. 3.3.4. Possible reverse transition to laminar flow Fig. 14 summarizes the axial velocity output signal obtained for a range of Reynolds number and different acceleration patterns. These results indicate that a transitional steady flow with Re in the range 4190–5930 can be re-laminarized by first applying a constant deceleration of a = 2.8 m/s2 followed by a constant acceleration greater than its critical value. Further experimental investigation is desirable and necessary to clarify if constant deceleration can similarly induce reverse transition from turbulent flow to laminar flow. 4. Conclusions Hot-wire experiments were performed in this study to clarify whether the effect of deceleration or viscosity, predominates in constant-deceleration pipe flow. Five experimental conditions were considered with steady state Reynolds numbers varying from 4190 to 13,400. The main findings of the study on the velocity distribution and transition to turbulence can be summarized as follows:
i. The effect of viscous force on flow transition is more dominant than that of constant deceleration when the Reynolds number in the initial steady flow is in the range 4190 to 5930. This implies that in this regime, a steady transitional flow becomes laminar at the constant-deceleration stage. ii. Laminar constant-deceleration flow is adequately predicted by numerical simulation. iii. Turbulent quasi-steady state occurs in the subsequent constant deceleration stage when the Reynolds number in the initial steady flow exceeds 11,000. iv. Viscous force suppresses the generation of turbulence against deceleration effect. This result, coupled with previous findings on transition to turbulence in constantacceleration pipe flow suggests that the frictional loss in pipe flows can be reduced by imposing unsteadiness. An example is the imposition of periodic constant acceleration and deceleration on steady pipe flows, as shown in Fig. 14.
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