Pulsating pipe flow instability perturbed by a trip wire
1.1 On flows over a circular cylinder in the critical transition regime
1.1.1 General characteristics of flow
The general characteristics of flow over a circular cylinder in the critical transition regime was studied by Miau et al.
[18]. The distribution curves of 𝐶𝐶𝑝𝑝𝑝𝑝 versus the Reynolds number are reproduced in Fig. 4 for discussion. The quantity of 𝐶𝐶𝑝𝑝𝑝𝑝 denotes the base pressure coefficient reduced from the time-mean pressure obtained at θ = 180o, where θ = 0o denotes the forward stagnation point of the circular cylinder. The expression of 𝐶𝐶𝑝𝑝𝑝𝑝 is given below.
𝐶𝐶𝑝𝑝𝑝𝑝� 𝑃𝑃180°1 �𝑃𝑃0
2𝜌𝜌𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟2 (1)
𝑃𝑃180° denotes the base pressure measured at θ =180o; 𝑃𝑃0
denotes the static pressure in the free stream measured by the Pitot tube; ρ denotes the density of the air; Vref is the reference velocity, reduced from the difference between the time-averaged pressure at θ = 0o and 𝑃𝑃0. In addition, the Reynolds number is based on Vref and the diameter of the circular cylinder, D.
Fig. 4 Base pressure coefficients of smooth circular cylinders versus the Reynolds numbers measured [18]
In the figure, the data of the acrylic and stainless steel cylinders marked with (present) were obtained from Miau et al. [18]; the distribution curves obtained from Bearman [1], ESDU [16] and Kao [19] were included for comparison. First of all, a trend in common revealed by all the distribution curves for the smooth circular cylinders is that a pronounced transition of 𝐶𝐶𝑝𝑝𝑝𝑝 takes place in the range of Re between 2×105 to 4×105, where 𝐶𝐶𝑝𝑝𝑝𝑝 is increased drastically from -1.2 to -0.4 or even less negative. Second, with a detailed look on these distributions one can find that they are not agreed perfectly, instead some scatterings among the distributions. The variations were attributed to a number of the influencing factors, including the wind tunnel freestream turbulence and the surface roughness of the circular cylinder [1, 20-23]. Third, it is interesting to mention that the pronounced transition is actually associated with a process of transition through different states. According to Zdravkovich [5], the critical regime consists of the one-bubble state and the two-bubble state. In the one-bubble state, owing to that LSB could be situated on either side of the circular cylinder, the aerodynamic flow would appear in an asymmetric distribution. As a result, significant lift was produced.
The intermediate states in the critical transition range can be illustrated by the oi-film flow visualization photos in Fig. 5, which were reproduced from Miau et al. [18]. Specifically, the photo in Fig. 5a unveils the surface oil-film pattern at Re = 3.09×105 called the pre-critical state that the flow separation line can be identified at θ = 100o, which is significantly delayed in comparison with that learned from the literature in the subcritical regime, which is known in the neighborhood of θ = 80o [24, 25]. As noted in Fig. 4, the 𝐶𝐶𝑝𝑝𝑝𝑝 value at Re =3.09×105 falls between 0.8 to -1, which is less negative than -1.2 of the subcritical state.
The photos in Fig. 5b are referred to the bi-stable one-bubble state, which was taken at Re = 3.64×105. On the θ
=-90o side, a clear indication of the presence of the bubble is shown, that the flow reattachment due to the separation bubble can be identified in the region of θ =-100o to -110o, nevertheless not uniformly aligned in the spanwise direction. This appearance infers that flow around the circular cylinder was strongly three-dimensional. On the θ
= 90o side, a separation line is rather identified upstream of θ = 90o, which is skewed with respect to the spanwise direction. The oil-film pattern also unveils a vague footprint of the separation bubble like that on the θ =-90o side, which is explained due to that during the experiment a separation bubble could also unsteadily present on this side.
In Fig. 5c, the photos taken at Re = 3.95×105 are referred to the stably two-bubble state, inferring that two separation bubbles were situated stably on two sides of the circular cylinder, respectively. Again, the signatures of the separation bubbles on the circular cylinder infer that the flow structures would be strongly three-dimensional.
2 might fail to reattach and became an unattached free shear layer. The hysteresis of bubble bursting on an airfoil due to changing the incidence caught significant attention in these studies.
It is known that flow in the critical regime is highly unsteady, even non-stationary, which is intimately linked with the development of the laminar separation bubble(s).
This review paper is focused on the experimental findings associated with the unsteadiness of flow due to the development of separation bubble(s) in the critical regime of the blunt models which have been studied in the past years. The blunt models described below include a two-dimensional circular cylinder and a teardrop shaped model.
On the blunt model of a two-dimensional circular cylinder, the experiments were carried out in the ABRI wind tunnel [15], which is a closed-loop design with two test sections. The present model was situated on a turntable centered at 2.9 m downstream of the inlet of the primary test section, which is 4 m (width) by 2.6 m (height) in cross-section and 36.5 m in length. The turbulence intensity and non-uniformity of the flow measured at the inlet of the primary test section were less than 0.3% and 0.4%, respectively, for a mean velocity up to 36 m/s.
There were two circular cylinders employed for experiment, one of which was made of acrylics, 300 mm in diameter, and the other was made of stainless steel, 320 mm in diameter. The geometric blockage ratios of the acrylics and stainless-steel cylinders were 7.5% and 8%, respectively.
Figure 1 shows the photos of the two models spanning the top and bottom walls in the test section. The averaged heights of roughness of the acrylic and stainless steel cylinders were 0.73 μm and 12.4 μm, respectively; the values of the corresponding relative roughness, based on the diameters of the cylinders, were 2.43×10-6 and 3.88×
10-5, respectively, which are regarded as the smooth circular cylinders [16].
(a) carried out in an open jet wind tunnel [17]. The length of the open-jet test section was 2.5 m, and the jet diameter was 0.5m. The freestream turbulent intensity measured at the upstream of the test section was 0.6%. The mean velocity of the incoming flow could reach above 35 m/s.
The teardrop model was manufactured by a 3D printer, on which the pressure taps were located on the model surface.
Shown in Fig. 2 is the contour profile of the teardrop surfaces at the section 0.4 span from the root. The model was 0.1 m in chord length, 0.04m at the maximum thickness and 0.4 m in span. Thus, the blockage ratio of the models was 8%. During the experiment, the critical transition range was identified for the Reynolds numbers in a range from 4×104 to 7×104, based on the chord length and the incoming velocity measured by a Pitot tube situate downstream of the convergent nozzle.
Fig. 2 The contour profile of the teardrop model [17]
― 4 ―
J.-J. MIAU, Y.-H. LAI, P. DONG and A. ZOGHLAMI
3 Fig. 3 A view of the model situated vertically in the test section of an open-jet wind tunnel [17]. The coordinate system employed is included for reference
1.1 On flows over a circular cylinder in the critical transition regime
1.1.1 General characteristics of flow
The general characteristics of flow over a circular cylinder in the critical transition regime was studied by Miau et al.
[18]. The distribution curves of 𝐶𝐶𝑝𝑝𝑝𝑝 versus the Reynolds number are reproduced in Fig. 4 for discussion. The quantity of 𝐶𝐶𝑝𝑝𝑝𝑝 denotes the base pressure coefficient reduced from the time-mean pressure obtained at θ = 180o, where θ = 0o denotes the forward stagnation point of the circular cylinder. The expression of 𝐶𝐶𝑝𝑝𝑝𝑝 is given below.
𝐶𝐶𝑝𝑝𝑝𝑝 �𝑃𝑃180°1 �𝑃𝑃0
2𝜌𝜌𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟2 (1)
𝑃𝑃180° denotes the base pressure measured at θ =180o; 𝑃𝑃0
denotes the static pressure in the free stream measured by the Pitot tube; ρ denotes the density of the air; Vref is the reference velocity, reduced from the difference between the time-averaged pressure at θ = 0o and 𝑃𝑃0. In addition, the Reynolds number is based on Vref and the diameter of the circular cylinder, D.
Fig. 4 Base pressure coefficients of smooth circular cylinders versus the Reynolds numbers measured [18]
In the figure, the data of the acrylic and stainless steel cylinders marked with (present) were obtained from Miau et al. [18]; the distribution curves obtained from Bearman [1], ESDU [16] and Kao [19] were included for comparison. First of all, a trend in common revealed by all the distribution curves for the smooth circular cylinders is that a pronounced transition of 𝐶𝐶𝑝𝑝𝑝𝑝 takes place in the range of Re between 2×105 to 4×105, where 𝐶𝐶𝑝𝑝𝑝𝑝 is increased drastically from -1.2 to -0.4 or even less negative. Second, with a detailed look on these distributions one can find that they are not agreed perfectly, instead some scatterings among the distributions. The variations were attributed to a number of the influencing factors, including the wind tunnel freestream turbulence and the surface roughness of the circular cylinder [1, 20-23]. Third, it is interesting to mention that the pronounced transition is actually associated with a process of transition through different states. According to Zdravkovich [5], the critical regime consists of the one-bubble state and the two-bubble state.
In the one-bubble state, owing to that LSB could be situated on either side of the circular cylinder, the aerodynamic flow would appear in an asymmetric distribution. As a result, significant lift was produced.
The intermediate states in the critical transition range can be illustrated by the oi-film flow visualization photos in Fig. 5, which were reproduced from Miau et al. [18].
Specifically, the photo in Fig. 5a unveils the surface oil-film pattern at Re = 3.09×105 called the pre-critical state that the flow separation line can be identified at θ = 100o, which is significantly delayed in comparison with that learned from the literature in the subcritical regime, which is known in the neighborhood of θ = 80o [24, 25]. As noted in Fig. 4, the 𝐶𝐶𝑝𝑝𝑝𝑝 value at Re =3.09×105 falls between 0.8 to -1, which is less negative than -1.2 of the subcritical state.
The photos in Fig. 5b are referred to the bi-stable one-bubble state, which was taken at Re = 3.64×105. On the θ
=-90o side, a clear indication of the presence of the bubble is shown, that the flow reattachment due to the separation bubble can be identified in the region of θ =-100o to -110o, nevertheless not uniformly aligned in the spanwise direction. This appearance infers that flow around the circular cylinder was strongly three-dimensional. On the θ
= 90o side, a separation line is rather identified upstream of θ = 90o, which is skewed with respect to the spanwise direction. The oil-film pattern also unveils a vague footprint of the separation bubble like that on the θ =-90o side, which is explained due to that during the experiment a separation bubble could also unsteadily present on this side.
In Fig. 5c, the photos taken at Re = 3.95×105 are referred to the stably two-bubble state, inferring that two separation bubbles were situated stably on two sides of the circular cylinder, respectively. Again, the signatures of the separation bubbles on the circular cylinder infer that the flow structures would be strongly three-dimensional.
2 might fail to reattach and became an unattached free shear layer. The hysteresis of bubble bursting on an airfoil due to changing the incidence caught significant attention in these studies.
It is known that flow in the critical regime is highly unsteady, even non-stationary, which is intimately linked with the development of the laminar separation bubble(s).
This review paper is focused on the experimental findings associated with the unsteadiness of flow due to the development of separation bubble(s) in the critical regime of the blunt models which have been studied in the past years. The blunt models described below include a two-dimensional circular cylinder and a teardrop shaped model.
On the blunt model of a two-dimensional circular cylinder, the experiments were carried out in the ABRI wind tunnel [15], which is a closed-loop design with two test sections. The present model was situated on a turntable centered at 2.9 m downstream of the inlet of the primary test section, which is 4 m (width) by 2.6 m (height) in cross-section and 36.5 m in length. The turbulence intensity and non-uniformity of the flow measured at the inlet of the primary test section were less than 0.3% and 0.4%, respectively, for a mean velocity up to 36 m/s.
There were two circular cylinders employed for experiment, one of which was made of acrylics, 300 mm in diameter, and the other was made of stainless steel, 320 mm in diameter. The geometric blockage ratios of the acrylics and stainless-steel cylinders were 7.5% and 8%, respectively.
Figure 1 shows the photos of the two models spanning the top and bottom walls in the test section. The averaged heights of roughness of the acrylic and stainless steel cylinders were 0.73 μm and 12.4 μm, respectively; the values of the corresponding relative roughness, based on the diameters of the cylinders, were 2.43×10-6 and 3.88×
10-5, respectively, which are regarded as the smooth circular cylinders [16].
(a) carried out in an open jet wind tunnel [17]. The length of the open-jet test section was 2.5 m, and the jet diameter was 0.5m. The freestream turbulent intensity measured at the upstream of the test section was 0.6%. The mean velocity of the incoming flow could reach above 35 m/s.
The teardrop model was manufactured by a 3D printer, on which the pressure taps were located on the model surface.
Shown in Fig. 2 is the contour profile of the teardrop surfaces at the section 0.4 span from the root. The model was 0.1 m in chord length, 0.04m at the maximum thickness and 0.4 m in span. Thus, the blockage ratio of the models was 8%. During the experiment, the critical transition range was identified for the Reynolds numbers in a range from 4×104 to 7×104, based on the chord length and the incoming velocity measured by a Pitot tube situate downstream of the convergent nozzle.
Fig. 2 The contour profile of the teardrop model [17]
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Advanced Experimental Mechanics, Vol.4 (2019)
5 However, little information concerning the fluctuating characteristics could be inferred by the oil-film flow visualization photo, because it provided the information of time-mean flow characteristics mainly.
1.1.2 Non-stationary and three-dimensional flow characteristics
The non-stationary and three-dimensional flow characteristics in the critical transition range, due to the formation of the separation bubbles on the circular cylinder, were studied in Lin et al. [26]. The experiments were carried out with the acrylic circular cylinder, on which the pressure taps were located along the spanwise and circumferential directions. Figure 8 shows a schematic drawing of the circular cylinder with the pressure taps. For more details of the experimental setup, one may refer to Lin et al. [26].
Fig. 8 A schematic drawing of the acrylic circular cylinder with the indication of the pressure taps [26]
Fig. 9 Real-time 𝐶𝐶𝑝𝑝 traces obtained at z = 0, ±0.5 and ±1 D, θ = 78o, for Re = 3.4×105 [26]
The real-time flow characteristics for Re = 3.4×105, in terms of the instantaneous 𝐶𝐶� traces are shown in Fig. separation bubble appeared at the spanwise locations of z = 1 D and 0.5 D, but not at z = 0, -1 D and -0.5 D.
It should be mentioned that the unsteady, three-dimensional characteristics might be difficult to be realized
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Fig. 5 Oil-film visualization photos of the stainless steel circular cylinder: (a) Re = 3.09×105, the pre-critical state, (b) Re = 3.64×105, the bi-stable one-bubble state, (c) Re = 3.95×105, the stably two-bubble state [18]
Further information regarding the flow distributions around the two circular cylinders in the critical transition range can be learned from Figs. 6 and 7, respectively. In both figures, the quantities of 𝐶𝐶𝑝𝑝 and 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 are defined as angular position θ on the cylinder surface, where θ = 0 to
±180o; Prms denotes the root-mean-square value of pressure fluctuations measured on the cylinder at θ.
Interesting features noted from the distributions of 𝐶𝐶𝑝𝑝 and
𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 in the two figures are described below. First, in the critical transition range the 𝐶𝐶𝑝𝑝 distributions for either of the circular cylinders show significant variations with the Reynolds numbers. This observation unveils the transitional flow behavior, in addition to the appearance of the 𝐶𝐶𝑝𝑝𝑝𝑝 distributions seen in Fig. 4. Second, in either of the figures the 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 distribution obtained at a pre-critical Reynolds number shows remarkably large values in the regions of θ = ±60° to ±90°. For instance, in Fig. 6 for the acrylic circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.10×105 reaches almost 0.4 at θ of about ±60°; in Fig. 7 for the stainless steel circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.08×105 reaches almost 0.8 at θ of about 80°. The pronounced pressure fluctuations are attributed to the transition of the states, like the situation of bifurcation of flow states noted by Schewe [3]. The present case is due to the transition from the subcritical to one-bubble states.
One might refer to that the oil-film flow visualization photo in Fig. 5a, which was obtained at the Reynolds number almost the same as those mentioned above.
(a)
(b)
Fig. 6 (a) Cp and (b) CPrms distributions of the acrylic circular cylinder. In (a), the Cp distributions obtained from ESDU [16] for Re = 3.10×105 and 3.46×105 are included
J.-J. MIAU, Y.-H. LAI, P. DONG and A. ZOGHLAMI
5 However, little information concerning the fluctuating characteristics could be inferred by the oil-film flow visualization photo, because it provided the information of time-mean flow characteristics mainly.
1.1.2 Non-stationary and three-dimensional flow characteristics
The non-stationary and three-dimensional flow characteristics in the critical transition range, due to the formation of the separation bubbles on the circular cylinder, were studied in Lin et al. [26]. The experiments were carried out with the acrylic circular cylinder, on which the pressure taps were located along the spanwise and circumferential directions. Figure 8 shows a schematic drawing of the circular cylinder with the pressure taps. For more details of the experimental setup, one may refer to Lin et al. [26].
Fig. 8 A schematic drawing of the acrylic circular cylinder with the indication of the pressure taps [26]
Fig. 9 Real-time 𝐶𝐶𝑝𝑝 traces obtained at z = 0, ±0.5 and ±1 D, θ = 78o, for Re = 3.4×105 [26]
The real-time flow characteristics for Re = 3.4×105, in terms of the instantaneous 𝐶𝐶� traces are shown in Fig. separation bubble appeared at the spanwise locations of z = 1 D and 0.5 D, but not at z = 0, -1 D and -0.5 D.
It should be mentioned that the unsteady, three-dimensional characteristics might be difficult to be realized
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Fig. 5 Oil-film visualization photos of the stainless steel circular cylinder: (a) Re = 3.09×105, the pre-critical state, (b) Re = 3.64×105, the bi-stable one-bubble state, (c) Re = 3.95×105, the stably two-bubble state [18]
Further information regarding the flow distributions around the two circular cylinders in the critical transition range can be learned from Figs. 6 and 7, respectively. In both figures, the quantities of 𝐶𝐶𝑝𝑝 and 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 are defined as angular position θ on the cylinder surface, where θ = 0 to
±180o; Prms denotes the root-mean-square value of pressure fluctuations measured on the cylinder at θ.
Interesting features noted from the distributions of 𝐶𝐶𝑝𝑝 and
𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 in the two figures are described below. First, in the critical transition range the 𝐶𝐶𝑝𝑝 distributions for either of the circular cylinders show significant variations with the Reynolds numbers. This observation unveils the transitional flow behavior, in addition to the appearance of the 𝐶𝐶𝑝𝑝𝑝𝑝 distributions seen in Fig. 4. Second, in either of the figures the 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 distribution obtained at a pre-critical Reynolds number shows remarkably large values in the regions of θ = ±60° to ±90°. For instance, in Fig. 6 for the acrylic circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.10×105 reaches almost 0.4 at θ of about ±60°; in Fig. 7 for the stainless steel circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.08×105 reaches almost 0.8 at θ of about 80°. The pronounced pressure fluctuations are attributed to the transition of the states, like the situation of bifurcation of flow states noted by Schewe [3]. The present case is due
𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 in the two figures are described below. First, in the critical transition range the 𝐶𝐶𝑝𝑝 distributions for either of the circular cylinders show significant variations with the Reynolds numbers. This observation unveils the transitional flow behavior, in addition to the appearance of the 𝐶𝐶𝑝𝑝𝑝𝑝 distributions seen in Fig. 4. Second, in either of the figures the 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 distribution obtained at a pre-critical Reynolds number shows remarkably large values in the regions of θ = ±60° to ±90°. For instance, in Fig. 6 for the acrylic circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.10×105 reaches almost 0.4 at θ of about ±60°; in Fig. 7 for the stainless steel circular cylinder, the maximum 𝐶𝐶𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 value in the distribution of Re = 3.08×105 reaches almost 0.8 at θ of about 80°. The pronounced pressure fluctuations are attributed to the transition of the states, like the situation of bifurcation of flow states noted by Schewe [3]. The present case is due