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Experimental investigation of the instability of spanwise-periodic low-speed streaks
Fluid Dynamics Research 34 (2004) 299 – 315
Experimental investigation of the instability of spanwise-periodic low-speed streaks Yasufumi Konishi, Masahito Asai∗ Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan Received 12 June 2003; received in revised form 19 December 2003; accepted 18 February 2004 Communicated by S. Kida
Abstract The streak instability is examined experimentally by arti3cially generating spanwise-periodic low-speed streaks in a laminar boundary layer on a 4at plate. Fundamental and subharmonic modes are excited for each of the sinuous and varicose instabilities and their development is compared with the corresponding result of a single low-speed streak. The development of subharmonic sinuous mode does not strongly depend on the streak spacing and it grows with almost the same growth rate as that for the single streak. By contrast, the development of fundamental sinuous mode is very sensitive to the streak spacing and is completely suppressed when the streak spacing is smaller than a critical value, about 2.5 times the streak width for the low-speed streaks examined. On the varicose instability, the fundamental mode is less ampli3ed than the subharmonic mode, but the growth of both modes is weak compared with the case of the single streak. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Instability; Low-speed streaks; Boundary layer transition; Wall turbulence
1. Introduction The near-wall streaky structure with high- and low-speed regions aligned in the streamwise direction is a typical coherent structure observed in wall turbulence as well as in laminar-turbulent transition of wall-bounded shear 4ows. When the near-wall streaks are intensi3ed, the associated three-dimensional velocity 3eld develops in4ectional velocity pro3les across the low-speed region, and thus the low-speed streaks may become highly unstable, generating energetic quasi-streamwise ∗
Corresponding author. Fax: +81-42-585-8652. E-mail address: [email protected] (M. Asai).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.4uiddyn.2004.02.003
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vortices. This instability is called the streak instability and is known as a key mechanism of generating and sustaining wall turbulence. In the present study, a well-controlled stability experiment is conducted on arti3cially generated low-speed streaks to show some important features of the streak instability. The instability of low-speed streaks was 3rst studied to understand the secondary instability process occurring after the growth of GEortler vortices on a concave wall. The GEortler vortices develop low-speed streaks with highly in4ectional velocity pro3les both in the normal-to-wall and spanwise directions due to each pair of counter-rotating streamwise vortices, and therefore the GEortler-vortex 4ow becomes linearly unstable to generate the so-called varicose and sinuous modes which develop hairpin-shaped (or arch-like) vortices and wavy meandering motions of GEortler vortices, respectively, as observed by Swearingen and Blackwelder (1987): Also see a review paper by Saric (1994). The stability analysis of the secondary instability was made by Hall and Horseman (1991), Yu and Liu (1991), Park and Huerre (1995), and Li and Malik (1995). The occurrence and subsequent instability of low-speed streaks were also observed in the boundary layer transition caused by high-intensity freestream turbulence. In such by-pass transition, the streak instability leading to the breakdown of low-speed streaks is responsible for the onset of wall turbulence. Matsubara and Alfredsson (2001) demonstrated experimentally that near-wall streaks were surely caused by free-stream turbulence when the turbulence level was a few percents of the free-stream velocity in terms of streamwise velocity 4uctuations. When the near-wall streaks are intensi3ed, a secondary instability occurs, leading to the subsequent breakdown into turbulent spots. Reddy et al. (1998) examined the threshold energy of disturbances for the streak breakdown resulting in transition in plane channel 4ows at subcritical Reynolds numbers on the basis of the stability theory. The related experiments by Elofsson et al. (1999) also showed that the occurrence of the secondary instability of low-speed streaks requires a rather large amplitude of velocity variation across the low-speed streaks. The transition process caused by streak instability in a 4at plate boundary layer was studied in detail numerically by Brandt and Henningson (2002). As for wall turbulence, Hamilton et al. (1995) proposed a regeneration cycle of the near-wall streamwise vortices and associated low-speed streaks through DNS of a minimal plane Couette 4ow unit with streamwise and spanwise periodic boundary conditions at the lowest critical Reynolds number for sustaining wall turbulence. In the regeneration cycle, the occurrence of near-wall streamwise vortices results from the instability of low-speed streaks leading to the meandering wavy motion of the streaks. Importance of the streak instability in the regeneration process of near-wall coherent structures was also demonstrated by JimIenez and Pinelli (1999) through DNS of a turbulent channel 4ow. Itano and Toh (2001) also found a similar traveling wave solution corresponding to a nonlinear saturation stage of the streak instability for a minimal channel 4ow. Kawahara and Kida (2001) deduced the regeneration cycle in a turbulent Couette 4ow through dynamical system approach. For low-speed streaks in fully developed wall turbulence, Jeong et al. (1997); Schoppa and Hussain (2000) studied the linear instability of low-speed streaks deduced from DNS data of wall turbulence. Their results indicate that viscous diKusion of the streak is signi3cant for the growth of the instability modes so that the disturbance amplitude is not so much ampli3ed. On the inviscid linear instability of in3nitely thin streak shear layer, see a recent paper by Kawahara et al. (2003). Schoppa and Hussain (2002) also claim that under high-intensity background turbulence such as in wall turbulence, a transient growth is more important than the linear instability for the breakdown of low-speed streaks.
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In order to examine the streak instability experimentally in detail, it is important to realize laminar low-speed streaks and to introduce well-controlled disturbances. Asai et al. (2000, 2002) examined the instability of a single low-speed streak generated arti3cially in a laminar boundary layer. In their experiment, a small piece of screen was set normal to the boundary-layer plate to produce a laminar single low-speed streak. In the screen-generating streak, longitudinal vortices developing immediately downstream of the screen were found to be very weak. By using such an arti3cially generated streak without streamwise vorticity, they could obtain the detailed stability characteristics for sinuous and varicose modes. In actual transitional and turbulent 4ows, however, there does not appear a single low-speed streak but an array of low-speed streaks with mean spacing; for instance about 100 wall unit in wall turbulence. So, in the present experimental study, spanwise-periodic low-speed streaks are generated in a laminar boundary layer by using a periodic array of screens set normal to the boundary-layer plate, as has been done in the single streak experiment. Comparisons of the development of instability modes are made between the periodic streaks and the single streak.
2. Experimental setup and procedure The whole experiment is conducted in a low turbulence wind tunnel of open jet type, with the test section of 400 × 400 mm2 . This facility is the same as that used in the experiment by Asai et al. (2000, 2002). As illustrated in Fig. 1, a boundary-layer plate, which is 10 mm thick and 1100 mm long, is set parallel to the oncoming uniform 4ow in the test section. Spanwise-periodic low-speed streaks are produced in the boundary layer by using small pieces of screens of 6 mm width and 2:5 mm height (or 3 mm height) set normal to the boundary-layer plate at a location 500 mm downstream of the leading edge. Each screen is a 40-mesh wire-gauze whose porosity
Fig. 1. Schematic of boundary layer plate (not to scale). s = 12; 15 or 18 mm.
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is 0.7. Seventeen screens are set with an equal interval of s in the spanwise direction. The spanwise spacing of the screens (or the streak spacing) s is selected to be 12 mm, 15 mm or 18 mm, which are 2, 2.5 and 3 times the width of each screen ls (=6 mm). As for the coordinate system, x is the streamwise distance measured from the leading edge, y the normal-to-wall distance, and z the spanwise distance. The x-location of the screens, x = 500 mm, is denoted by x0 . The free-stream velocity U∞ was 3xed at 4 m=s throughout the experiment. The free-stream turbulence is less than 0.1% of U∞ . Due to the low background turbulence, the laminar boundary layer develops without occurrence of any instability waves in the absence of the streak-generating screen. The velocity distributions without the streak-generating screens are well represented by Blasius 4ow pro3les. The displacement thickness of the Blasius boundary layer without the screens ∗B is 2:4 mm at x = x0 and U∞ = 4 m=s. Note that the height of the screen hs (=2:5 mm or 3:0 mm) is so chosen as to be close to ∗B . The Reynolds number based on ∗B and U∞ , i.e. R∗ = ∗B U∞ = (where is the kinematic viscosity) is about 650 at x = x0 . In order to excite varicose and sinuous instability modes separately, well-controlled sinusoidal disturbances are introduced into the laminar streak 4ow by alternately sucking and blowing air through small holes. The disturbances are given only to seven low-speed streaks (developing behind the seven central screens) among the total 17 low-speed streaks. To excite varicose modes, a hole of 3 mm in diameter is drilled just behind the midspan of each screen, 8:5 mm downstream of the screens as illustrated in Fig. 1. Each hole is connected to a loudspeaker by a vinyl hose. To excite sinuous modes, holes of 2 mm in diameter are drilled at both edges of each screen, 13:5 mm downstream of the screens, and are connected to two loudspeakers. The two loudspeakers are driven separately with sine-wave signals that are 180◦ out of phase. In addition, we can select two kinds of excitation for each of varicose and sinuous modes. One is in-phase excitation which can produce fundamental modes having the same phase relation for all the low-speed streaks. The other is anti-phase excitation that is 180◦ out of phase between the neighboring streaks, and it can introduce subharmonic modes. The fundamental modes have the same dominant spanwise wavelength as the streak spacing s , whereas the subharmonic modes have the spanwise wavelength twice the streak spacing 2s . The details of the excited disturbances will be explained later. A constant temperature hot-wire anemometer with a linearizer is used to measure time-mean and 4uctuation velocities in the streamwise direction, U and u. The sensitive length of hot-wire sensor, a tungsten wire of 5 m in diameter, is 1 mm. The hot-wire probe can be traversed in the x, y and z directions. The y-distributions of U and u (the r.m.s. value of u-4uctuation) are recorded in an XY plotter during each y-traverse. The hot-wire data are also stored in a personal computer.
3. Velocity eld downstream of the screens First let us explain the low-speed streaks arti3cially generated by the periodically mounted screens. Fig. 2(a) illustrates the y-distributions of time-mean velocity U at x − x0 = 50 mm for the low-speed streaks generated by the screens of (s ; hs ; ls ) = (12; 2:5; 6 mm). The y-distribution at midspan of each screen is well approximated by a tanh(y) pro3le, whereas the y-distribution between the neighboring screens is almost the same as the Blasius pro3le. Fig. 2(b) illustrates the z-distributions of U measured at several heights, y =1; 2; 3; 4 and 5 mm, at the same x-location. A velocity defect
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Fig. 2. The y- and z-distributions of mean velocity U at x − x0 = 50 mm. (s ; hs ; ls ) = (12; 2:5; 6 mm). (a) y-distributions at z = 6 mm ( ) and z = 0 (), (b) z-distributions at y = 5; 4; 3; 2 and 1 mm.
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due to the drag of the screens is observed markedly below y = 4 mm, and the spanwise velocity distributions across each low-speed streak are not unlike that of a usual wake. The lateral scale of each low-speed streak which is de3ned as the half-value width used for usual wakes is almost the same as (or only slightly smaller than) the width of each screen (ls = 6 mm). We also notice that the velocity excess induced at the side edges of the screen by the longitudinal (stationary horseshoe) vortices is small. This indicates that the eKect of the stationary horseshoe vortex on the velocity 3eld is really weak. Fig. 3 illustrates the streamwise development of low-speed streaks in terms of an iso-velocity contours in the x–z plane at y = 3 mm. The low-speed streaks extend downstream beyond x − x0 = 200 mm (200 mm downstream from the screens) though the velocity defect across each low-speed streak 3lls up gradually with x owing to the momentum transfer by viscous stresses in the lateral and normal-to-wall directions. The slow diKusion of the streak shear layers is due to the fact that the low-speed streaks remain laminar far downstream. The r.m.s. value of u-4uctuations is less than 1% of U∞ even at x − x0 = 200 mm, as shown in Fig. 4 which illustrates the y–z
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Fig. 3. Low-speed streaks downstream of the screens with (s ; hs ; ls )=(12; 2:5; 6 mm) represented by iso-velocity contours in x–z plane at y = 3 mm. Contour levels; U=U∞ = 0:3–0.7.
Fig. 4. The y- and z-distributions of r.m.s. value of u-4uctuations at x − x0 = 200 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm). Contour levels; u1 =U∞ = 0:001–0.005.
distributions of the r.m.s. Value u at x − x0 = 200 mm. The distribution of u takes maxima on the left and right vertical (@U=@z) shear layers and almost vanishes at the midspan z = 0, which indicates the feature of sinuous instability modes. The distribution of sinuous mode will be presented later. Similar periodic low-speed streaks are realized for the screen arrangement with larger spacing s . Fig. 5 compares in three cases of low-speed streaks with s = 12; 15 and 18 mm by iso-velocity contours in the y–z cross-section at x − x0 = 60 mm. The velocity 3eld developing downstream of each screen is almost the same in all the cases. We also con3rmed that the y-distributions at midspan of each screen in these three cases are almost completely the same.
4. Excitation of instability modes The in4ectional velocity pro3les across each low-speed streak are unstable to varicose and sinuous modes. Under background freestream turbulence, this instability nature may amplify ‘natural’ disturbances. In the present low-speed streaks, however, the 4ow remains laminar completely in the
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Fig. 5. Iso-velocity contours in y–z plane at x − x0 = 60 mm. Contour levels; U=U∞ = 0:1–0.9. (a) s = 12 mm, (b) s = 15 mm, (c) s = 18 mm. (hs ; ls ) = (2:5; 6 mm).
whole observation region up to x − x0 = 200 mm owing to low background turbulence less than is only 0.1% of U∞ . As already shown in Fig. 4, the maximum r.m.s. value of u-4uctuations um about 1% of U∞ even at x − x0 = 200 mm. This fact enables us to examine the streak instability by introducing arti3cial disturbances. Unlike the case of a single low-speed streak, interactions between disturbances developing along each low-speed streak might have a signi3cant eKect on the instability in the periodic low-speed streaks. So, we introduced arti3cial disturbances in two diKerent ways to excite fundamental and subharmonic modes for each of the varicose and sinuous instabilities. For the fundamental varicose modes, sinusoidal disturbances given to all the low-speed streaks are in phase, and therefore the excited varicose modes have the spanwise periodicity of the same dominant wavelength s as the low-speed streaks. For the subharmonic varicose modes, on the other hand, the disturbances are 180◦ out of phase between the neighboring streaks and they have the periodicity of twice the streak spacing 2s . These are also the cases for excitation of sinuous modes. The in-phase forcing to all the low-speed streaks excites fundamental sinuous modes having the dominant wavelength s , and the anti-phase forcing to the neighboring low-speed streaks excites subharmonic sinuous modes having the dominant wavelength 2s . Figs. 6(a) and (b), respectively, illustrate the fundamental and subharmonic varicose modes excited in the streaks with the spacing s = 12 mm in terms of instantaneous streamwise velocity 4uctuation u1 (the forcing frequency component of u-4uctuation) in the x–z plane at y=3 mm. Here, the forcing frequency is 90 Hz, and these instantaneous velocity 4uctuation 3elds were obtained by means of phase averaging technique. The disturbances along all the low-speed streaks are in-phase for the fundamental mode, as seen in Fig. 6(a). On the other hand, subharmonic varicose modes have the dominant wavelength of 2s , and therefore we can see a staggered pattern of iso-velocity contours in
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Fig. 6. Instantaneous u-4uctuation 3eld of varicose mode (f = 90 Hz) in the x–z plane at y = 3 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm), represented by iso-velocity lines of u1 =U∞ = −0:01–0.01. (a) Fundamental mode, (b) subharmonic mode.
Fig. 7. Amplitude distributions of varicose modes (f = 90 Hz) in the y–z plane at x − x0 = 60 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm). (a) Fundamental mode, represented by iso-intensity lines of u1 =U∞ = 0:0005–0.0045, (b) subharmonic mode, represented by iso-intensity lines of u1 =U∞ = 0:0006–0.0054.
Fig. 6(b). Note that the staggered pattern in Fig. 6(b) is not unlike that caused by a pair of oblique Tollmien–Schlichting waves which might be expected to appear if the low-speed streaks were so weak that the 4ow was considered to be Blasius 4ow. Figs. 7(a) and (b) illustrate the amplitude distributions of the fundamental and subharmonic varicose modes, respectively, in terms of u1 (the r.m.s. value of u1 ) in the y–z cross-section at x − x0 = 60 mm. Both of the amplitude distributions
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Fig. 8. Instantaneous u-4uctuation 3eld of sinuous mode (f = 60 Hz) in the x–z plane at y = 2:5 mm for (s ; hs ; ls ) = (18; 2:5; 6 mm). (a) Fundamental mode, represented by iso-velocity lines of u1 =U∞ = −0:003–0.003, (b) subharmonic mode, represented by iso-velocity lines of u1 =U∞ = −0:01–0.01.
are not so diKerent from each other except in the high-speed region between the neighboring two screens: The amplitude u1 becomes zero at z = ±6 mm for the subharmonic mode because of the 180◦ phase jump between the neighboring low-speed streaks, while it does not for the fundamental mode. Similarly, Figs. 8(a) and (b), respectively, show the fundamental and subharmonic sinuous modes excited in the streaks with s = 18 mm in terms of instantaneous velocity 4uctuations of forcing frequency (60 Hz) component u1 in the x–z plane at y = 2:5 mm. Superposition of u1 on the base streak 4ow U leads to the wavy structure of the low-speed streaks (or meandering motion of the streaks) in both the cases. Figs. 9(a) and (b) illustrate, respectively, the corresponding amplitude distributions u1 (y; z) of the fundamental and subharmonic sinuous modes at x−x0 =60 mm, showing that both the amplitude distributions across each low-speed streak are not so largely diKerent from each other. It will be seen later that such a slight diKerence in the structure causes a signi3cant diKerence in the disturbance growth. In the following, the development of these sinuous and varicose modes is examined in detail. 5. Development of sinuous modes Fig. 10(a) shows the streamwise development of subharmonic disturbances in the low-speed streaks with the smallest spacing s = 12 mm by plotting u1 m , the maximum amplitude of u1 against x. The forcing frequency is varied from 30 to 90 Hz. The disturbances can grow except 90 Hz. In particular,
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Fig. 9. Amplitude distributions of sinuous modes (f = 60 Hz) in the y–z plane at x − x0 = 60 mm. (s ; hs ; ls ) = (18; 2:5; 6 mm). (a) Fundamental mode, represented by iso-velocity lines of u1 =U∞ = 0:0002–0.0014, (b) subharmonic mode, represented by iso-velocity lines of u1 =U∞ = 0:0003–0.0027.
Fig. 10. Development of subharmonic sinuous modes. (a) s = 12 mm, (b) s = 18 mm. (hs ; ls ) = (2:5; 6 mm): ; 30; ; 45; ; 60; ; 70; O; 80; 5; 90 Hz.
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Fig. 11. Comparison of development of subharmonic sinuous modes excited at f = 60 Hz. (hs ; ls ) = (2:5; 6 mm). ; s = 12 mm; ; s = 15 mm; , s = 18 mm; , single low-speed streak.
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the disturbances of 45 and 60 Hz are ampli3ed about 20 times over the region from x − x0 = 30– 200 mm, and further continue to grow beyond x−x0 =200 mm. We also notice that the growth rate is decreasing gradually. This is because the low-speed streaks become weak in the downstream direction due to viscous diKusion of the associated vertical and horizontal shear layers. Fig. 10(b) illustrates the corresponding result for the low-speed streaks with the spacing s = 18 mm. The frequency of the most ampli3ed disturbance is the same between both the streaks of s = 12 and 18 mm, and the growth rate for s =18 mm is only slightly larger than that for s =12 mm. To show this feature more directly, Fig. 11 compares the streamwise development of the most ampli3ed disturbance (60 Hz) for the three cases of s = 12; 15 and 18 mm. The 3gure also includes the corresponding result for the single low-speed streak generated by using the same-sized screen (hs = 2:5 mm, ls = 6 mm), for comparison. It can be seen that the growth rate soon tends to the value for the single streak with increasing the streak spacing s . The growth rate for s = 18 mm already coincides with that for the single streak. Thus, the constraint of the periodic boundary condition little aKects on the streamwise growth of subharmonic modes. On the other hand, Fig. 12(a) illustrates the development of fundamental sinuous modes for the streak spacing s = 18 mm. The frequency of the most ampli3ed disturbance is around 60 Hz, not diKerent from that of the subharmonic mode. However, the maximum growth rate, estimated by d[ln u1 m ]=d x over x −x0 =50–100 mm, is only one-third that of the subharmonic mode. Furthermore, when the streak spacing s is decreased to 15 mm which gives s =ls = 2:5, the fundamental sinuous modes cease to grow. Fig. 12(b) illustrates the streamwise development of fundamental modes excited at various frequencies for s = 15 mm, showing that the disturbance development is nearly neutral. We also examined the development of fundamental modes in the case of s = 12 mm and con3rmed that there did not exist any growing fundamental disturbances for s = 12 mm. The large diKerence in the disturbance growth between the fundamental and subharmonic modes is attributed to slight diKerences in the disturbance structure. Comparing the u1 -3elds shown in Figs. 9(a) and (b), we can see that the fundamental mode interacts more strongly with disturbances along the neighboring low-speed streaks. As for the subharmonic mode, on the other hand, disturbance
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Fig. 12. Development of fundamental sinuous modes. (a) s = 18 mm, (b) s = 15 mm. (hs ; ls ) = (2:5; 6 mm). ; 45; ; 60; ; 70; O; 80; H; 90 Hz.
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along each low-speed streak seems to behave almost independently, which supports that the growth rate of the subharmonic mode is almost the same as that for the single low-speed streak. Since the sinuous instability is governed by the wake-type velocity distribution across each low-speed streak U (z), the growth rate might depend on the streak height relative to the streak width: The streak height can be de3ned as the height of the in4ection point of the velocity pro3le U (y). In this concern, Schoppa and Hussain (2002) analyzed the linear stability of the near-wall streaks deduced from DNS of wall turbulence and showed that the sinuous instability to fundamental modes could occur only for strong low-speed streaks with wall inclination angle of the streak vortex line above 50◦ . The large inclination angle corresponds to suSciently lifted-up low-speed streaks, so it is reasonable that the critical condition is related to the inclination angle for the streaks with similar geometry. In the present low-speed streaks generated by rectangular screens with the height-to-width ratio hs =ls = 0:42, the inclination angle is about 45◦ or less around x − x0 = 60 mm. If hs =ls is increased, the present low-speed steaks might turn to be unstable even for s =ls = 2:5 or less, though the streak geometry is not the same as that analyzed by Schoppa and Hussain (2002).
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Fig. 13. Iso-velocity lines in y–z plane at x − x0 = 60 mm. (s ; hs ; ls ) = (15; 3; 6 mm). Contour levels; U=U = 0:1–0.9.
In order to con3rm this, we examined the characteristics of sinuous instability by using 1.2 times taller screens with the screen width unchanged, i.e., (hs ; ls ) = (3 mm; 6 mm), which gives hs =ls = 0:5. Fig. 13 shows the velocity distribution in the y–z cross-section at x − x0 = 60 mm for the streaks generated by the screens of hs =ls = 0:5, with spacing s = 15 mm (s =ls = 2:5). Comparing with Fig. 5, it can be understood that the low-speed streaks extend to higher y-position. Therefore it is expected that the low-speed streaks becomes more unstable. The fundamental and subharmonic sinuous modes are excited in this case, and their streamwise development is shown in Figs. 14(a) and (b), respectively. As expected, the subharmonic mode rapidly grows with larger growth rate in this case, compared with the case of hs =ls = 0:42. The growth rate (d[ln u1 m ]=d x) of the most ampli3ed disturbance (60–70 Hz) estimated from the data over x − x0 = 50–100 mm is 0:038 [mm−1 ] for hs =ls = 0:50, while the corresponding growth rate is 0.023 [mm−1 ] for hs =ls = 0:42. The increase in hs =ls also operates to destabilize the low-speed streaks to fundamental modes, as seen in Fig. 14(a), which shows that disturbances of 60 and 70 Hz grow. However, the growth rates are very small, of the order of 0:01 [mm−1 ] and therefore the increase in hs =ls reduces only slightly the critical spacing s =ls for the growth of fundamental mode. Indeed, we con3rmed that the smallest streak spacing of s = 12 mm (s =ls = 2) suppressed the growth of fundamental sinuous modes completely in the case of hs =ls = 0:5, too. Thus, we may say that the critical streak spacing for the growth of fundamental sinuous mode is 2.5 or slightly less in terms of s =ls .
6. Development of varicose modes Figs. 15(a) and (b) illustrate the streamwise development of fundamental and subharmonic varicose modes in the case of s =12 mm (s =ls =2). Both the modes rapidly grow up to x −x0 =70 mm even for this smallest spacing s =ls = 2. The frequency of the most ampli3ed disturbance is at and around 90 Hz in both the cases, which is higher than the frequency of the most ampli3ed sinuous mode. The growth of varicose modes is essentially dominated by the Kelvin–Helmholts-type instability of the in4ectional velocity pro3le U (y), but their development is strongly aKected by the ratio of the streak width to the shear layer thickness. The disturbances undergo large ampli3cation for x − x0 ¡ 60 mm where the shear layer is thin compared with the streak width. As the streak shear layer is diKused by viscous diKusion, the growth rate is gradually reduced with increasing x and 3nally the disturbances cease to grow beyond x − x0 = 80 mm both for the fundamental and subharmonic modes.
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Fig. 14. Development of (a) fundamental and (b) subharmonic sinuous modes. (s ; hs ; ls ) = (15; 3; 6 mm). ; 45; ; 60; ; 70; O; 80; H; 90 Hz.
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In order to compare the ampli3cation of these fundamental and subharmonic varicose modes quantitatively, the N -factor de3ned as N = ln[u1 m (x)=u1 m (x1 )] is calculated for each frequency f. Here, x1 is selected to be the location 30 mm downstream of the screens, i.e., x1 − x0 = 30 mm, where the exponential growth has already started. The maximum value of N , which is denoted by Nmax , is obtained at an x-station where the amplitude u1 m attains a maximum for each frequency. Fig. 16 plots Nmax against the forcing frequency f. The 3gure includes the result of the single low-speed streak generated by the same-sized screen, i.e., ls = 6 mm and hs = 2:5 mm. Comparing the Nmax in the three cases, we see that both the fundamental and subharmonic modes are markedly suppressed due to the constraint of the spanwise periodicity imposed on the disturbances for all the frequencies examined. On the most ampli3ed disturbance, the values of Nmax for the fundamental and subharmonic modes are respectively about 50% and 65% smaller than that for the single streak. It is also worth noting that the reduction in the growth rate is more pronounced for higher-frequency disturbances (or smaller-scale disturbances). These results indicate that the three-dimensional nature of the instability is not neglected even though the growth of varicose modes is essentially governed by Kelvin–Helmholts instability of horizontal (@U=@y) shear layer.
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Fig. 15. Development of (a) fundamental and (b) subharmonic varicose modes. (s ; hs ; ls ) = (12; 2:5; 6 mm). ; 60; ; 75; H; 90; 4; 110; O; 130 Hz.
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Fig. 16. Nmax versus frequency f for varicose modes. (s ; hs ; ls )=(12; 2:5; 6 mm). mode; , single low-speed streak.
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; 45;
◦, fundamental mode; , subharmonic
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7. Conclusions The instability of spanwise-periodic low-speed streaks was studied experimentally. The structure of the low-speed streaks was well controlled by the lateral spacing (s ) of small pieces of screens as well as their lateral and vertical dimensions (ls and hs ). The low-speed streaks remained laminar far downstream in all the cases examined, which enabled us to examine the linear instability to both sinuous and varicose modes by introducing arti3cial disturbances. The fundamental and subharmonic disturbances which had dominant spanwise wavelenght of s and 2s , respectively, were also excited successfully for each of the sinuous and varicose instabilities. The main results are summarized below. On the sinuous instability, the subharmonic mode can grow with almost the same growth rate as that for the single low-speed streak. When the streak spacing s is three times larger than the streak width ls , the maximum growth rate is almost the same as that for the single low-speed streak. Even for s =ls = 2:0, the growth rate is only slightly smaller than that for the single streak. However, this is not the case for the fundamental sinuous modes. The constraint due to the spanwise periodicity inhibits the ampli3cation of fundamental sinuous modes with dominant wavelength of s when the s =ls is less than a critical value, though the critical streak spacing is weakly dependent on the height-to-width ratio hs =ls . The critical value of s =ls for the growth of fundamental sinuous modes is 2.5 for hs =ls = 0:42 and slightly less for hs =ls = 0:5. Even for larger streak spacing s =ls = 3 with hs =ls = 0:42, the spatial growth rate of the most ampli3ed fundamental mode is only about one-third that of the subharmonic mode. On the varicose instability, both the fundamental and subharmonic modes are less ampli3ed, compared with the case of single low-speed streak. The reduction in the ampli3cation is more pronounced for higher-frequency disturbances. Besides, the fundamental varicose modes are more largely aKected by the periodic constraint than the subharmonic varicose modes. Acknowledgements This work was in part supported by the Grant-in-Aid for Scienti3c Research C (No. 13650963) from the Japan Society for the Promotion of Science and the Grant-in-Aid for Special Scienti3c Research (No. 12125203) from the Ministry of Education, Sports, Culture, Science and Technology, Japan. References Asai, M., Minagawa, M., Nishioka, M., 2000. Instability and breakdown of the three-dimensional high-shear layer associated with a near-wall low-speed streak. In: Fasel, H., Saric, W. (Eds.), Laminar-Turbulent Transition. Springer, Berlin, pp. 269–274. Asai, M., Minagawa, M., Nishioka, M., 2002. The instability and breakdown of a near-wall low-speed streak. J. Fluid Mech. 455, 289–314. Brandt, L., Henningson, D.S., 2002. Transition of streamwise streaks in zero-pressure-gradient boundary layers. J. Fluid Mech. 472, 229–261. Elofsson, P.A., Kawakami, M., Alfredsson, P.H., 1999. Experiments on the stability of streamwise streaks in plane Poiseuille 4ow. Phys. Fluids 11, 915–930.
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Hall, P., Horseman, N.J., 1991. The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375. Hamilton, J.M., Kim, J., WaleKe, F., 1995. Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348. Itano, T., Toh, S., 2001. The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716. Jeong, J., Hussain, F., Schoppa, W., Kim, J., 1997. Coherent structures near the wall in a turbulent channel 4ow. J. Fluid Mech. 332, 185–214. JimIenez, J., Pinelli, A., 1999. The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359. Kawahara, G., Kida, S., 2001. Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300. Kawahara, G., JimIenez, J., Uhlmann, M., Pinelli, A., 2003. Linear instability of a corrugated vortex sheet—a model for streak instability. J. Fluid Mech. 483, 315–342. Li, F., Malik, M.R., 1995. Fundamental and subharmonic secondary instabilities of GEortler vortices. J. Fluid Mech. 297, 77–100. Matsubara, M., Alfredsson, P.H., 2001. Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168. Park, D.S., Huerre, P., 1995. Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate. J. Fluid Mech. 283, 249–272. Reddy, S.C., Schmid, P.J., Baggett, J.S., Henningson, D.S., 1998. On the stability of streamwise streaks and transition thresholds in plane channel 4ows. J. Fluid Mech. 365, 269–303. Saric, W.S., 1994. GEortler vortices. Annu. Rev. Fluid Mech. 26, 379–409. Schoppa, W., Hussain, F., 2000. Coherent structure dynamics in near-wall turbulence. Fluid Dyn. Res. 26, 119–139. Schoppa, W., Hussain, F., 2002. Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108. Swearingen, J.D., Blackwelder, R.F., 1987. The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290. Yu, X., Liu, J.T.C., 1991. The secondary instability in GEortler 4ow. Phys. Fluids A 3, 1845–1847.
Experimental investigation of the instability of spanwise-periodic low-speed streaks Yasufumi Konishi, Masahito Asai∗ Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan Received 12 June 2003; received in revised form 19 December 2003; accepted 18 February 2004 Communicated by S. Kida
Abstract The streak instability is examined experimentally by arti3cially generating spanwise-periodic low-speed streaks in a laminar boundary layer on a 4at plate. Fundamental and subharmonic modes are excited for each of the sinuous and varicose instabilities and their development is compared with the corresponding result of a single low-speed streak. The development of subharmonic sinuous mode does not strongly depend on the streak spacing and it grows with almost the same growth rate as that for the single streak. By contrast, the development of fundamental sinuous mode is very sensitive to the streak spacing and is completely suppressed when the streak spacing is smaller than a critical value, about 2.5 times the streak width for the low-speed streaks examined. On the varicose instability, the fundamental mode is less ampli3ed than the subharmonic mode, but the growth of both modes is weak compared with the case of the single streak. c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. Keywords: Instability; Low-speed streaks; Boundary layer transition; Wall turbulence
1. Introduction The near-wall streaky structure with high- and low-speed regions aligned in the streamwise direction is a typical coherent structure observed in wall turbulence as well as in laminar-turbulent transition of wall-bounded shear 4ows. When the near-wall streaks are intensi3ed, the associated three-dimensional velocity 3eld develops in4ectional velocity pro3les across the low-speed region, and thus the low-speed streaks may become highly unstable, generating energetic quasi-streamwise ∗
Corresponding author. Fax: +81-42-585-8652. E-mail address: [email protected] (M. Asai).
c 2004 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. 0169-5983/$30.00 All rights reserved. doi:10.1016/j.4uiddyn.2004.02.003
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vortices. This instability is called the streak instability and is known as a key mechanism of generating and sustaining wall turbulence. In the present study, a well-controlled stability experiment is conducted on arti3cially generated low-speed streaks to show some important features of the streak instability. The instability of low-speed streaks was 3rst studied to understand the secondary instability process occurring after the growth of GEortler vortices on a concave wall. The GEortler vortices develop low-speed streaks with highly in4ectional velocity pro3les both in the normal-to-wall and spanwise directions due to each pair of counter-rotating streamwise vortices, and therefore the GEortler-vortex 4ow becomes linearly unstable to generate the so-called varicose and sinuous modes which develop hairpin-shaped (or arch-like) vortices and wavy meandering motions of GEortler vortices, respectively, as observed by Swearingen and Blackwelder (1987): Also see a review paper by Saric (1994). The stability analysis of the secondary instability was made by Hall and Horseman (1991), Yu and Liu (1991), Park and Huerre (1995), and Li and Malik (1995). The occurrence and subsequent instability of low-speed streaks were also observed in the boundary layer transition caused by high-intensity freestream turbulence. In such by-pass transition, the streak instability leading to the breakdown of low-speed streaks is responsible for the onset of wall turbulence. Matsubara and Alfredsson (2001) demonstrated experimentally that near-wall streaks were surely caused by free-stream turbulence when the turbulence level was a few percents of the free-stream velocity in terms of streamwise velocity 4uctuations. When the near-wall streaks are intensi3ed, a secondary instability occurs, leading to the subsequent breakdown into turbulent spots. Reddy et al. (1998) examined the threshold energy of disturbances for the streak breakdown resulting in transition in plane channel 4ows at subcritical Reynolds numbers on the basis of the stability theory. The related experiments by Elofsson et al. (1999) also showed that the occurrence of the secondary instability of low-speed streaks requires a rather large amplitude of velocity variation across the low-speed streaks. The transition process caused by streak instability in a 4at plate boundary layer was studied in detail numerically by Brandt and Henningson (2002). As for wall turbulence, Hamilton et al. (1995) proposed a regeneration cycle of the near-wall streamwise vortices and associated low-speed streaks through DNS of a minimal plane Couette 4ow unit with streamwise and spanwise periodic boundary conditions at the lowest critical Reynolds number for sustaining wall turbulence. In the regeneration cycle, the occurrence of near-wall streamwise vortices results from the instability of low-speed streaks leading to the meandering wavy motion of the streaks. Importance of the streak instability in the regeneration process of near-wall coherent structures was also demonstrated by JimIenez and Pinelli (1999) through DNS of a turbulent channel 4ow. Itano and Toh (2001) also found a similar traveling wave solution corresponding to a nonlinear saturation stage of the streak instability for a minimal channel 4ow. Kawahara and Kida (2001) deduced the regeneration cycle in a turbulent Couette 4ow through dynamical system approach. For low-speed streaks in fully developed wall turbulence, Jeong et al. (1997); Schoppa and Hussain (2000) studied the linear instability of low-speed streaks deduced from DNS data of wall turbulence. Their results indicate that viscous diKusion of the streak is signi3cant for the growth of the instability modes so that the disturbance amplitude is not so much ampli3ed. On the inviscid linear instability of in3nitely thin streak shear layer, see a recent paper by Kawahara et al. (2003). Schoppa and Hussain (2002) also claim that under high-intensity background turbulence such as in wall turbulence, a transient growth is more important than the linear instability for the breakdown of low-speed streaks.
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In order to examine the streak instability experimentally in detail, it is important to realize laminar low-speed streaks and to introduce well-controlled disturbances. Asai et al. (2000, 2002) examined the instability of a single low-speed streak generated arti3cially in a laminar boundary layer. In their experiment, a small piece of screen was set normal to the boundary-layer plate to produce a laminar single low-speed streak. In the screen-generating streak, longitudinal vortices developing immediately downstream of the screen were found to be very weak. By using such an arti3cially generated streak without streamwise vorticity, they could obtain the detailed stability characteristics for sinuous and varicose modes. In actual transitional and turbulent 4ows, however, there does not appear a single low-speed streak but an array of low-speed streaks with mean spacing; for instance about 100 wall unit in wall turbulence. So, in the present experimental study, spanwise-periodic low-speed streaks are generated in a laminar boundary layer by using a periodic array of screens set normal to the boundary-layer plate, as has been done in the single streak experiment. Comparisons of the development of instability modes are made between the periodic streaks and the single streak.
2. Experimental setup and procedure The whole experiment is conducted in a low turbulence wind tunnel of open jet type, with the test section of 400 × 400 mm2 . This facility is the same as that used in the experiment by Asai et al. (2000, 2002). As illustrated in Fig. 1, a boundary-layer plate, which is 10 mm thick and 1100 mm long, is set parallel to the oncoming uniform 4ow in the test section. Spanwise-periodic low-speed streaks are produced in the boundary layer by using small pieces of screens of 6 mm width and 2:5 mm height (or 3 mm height) set normal to the boundary-layer plate at a location 500 mm downstream of the leading edge. Each screen is a 40-mesh wire-gauze whose porosity
Fig. 1. Schematic of boundary layer plate (not to scale). s = 12; 15 or 18 mm.
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is 0.7. Seventeen screens are set with an equal interval of s in the spanwise direction. The spanwise spacing of the screens (or the streak spacing) s is selected to be 12 mm, 15 mm or 18 mm, which are 2, 2.5 and 3 times the width of each screen ls (=6 mm). As for the coordinate system, x is the streamwise distance measured from the leading edge, y the normal-to-wall distance, and z the spanwise distance. The x-location of the screens, x = 500 mm, is denoted by x0 . The free-stream velocity U∞ was 3xed at 4 m=s throughout the experiment. The free-stream turbulence is less than 0.1% of U∞ . Due to the low background turbulence, the laminar boundary layer develops without occurrence of any instability waves in the absence of the streak-generating screen. The velocity distributions without the streak-generating screens are well represented by Blasius 4ow pro3les. The displacement thickness of the Blasius boundary layer without the screens ∗B is 2:4 mm at x = x0 and U∞ = 4 m=s. Note that the height of the screen hs (=2:5 mm or 3:0 mm) is so chosen as to be close to ∗B . The Reynolds number based on ∗B and U∞ , i.e. R∗ = ∗B U∞ = (where is the kinematic viscosity) is about 650 at x = x0 . In order to excite varicose and sinuous instability modes separately, well-controlled sinusoidal disturbances are introduced into the laminar streak 4ow by alternately sucking and blowing air through small holes. The disturbances are given only to seven low-speed streaks (developing behind the seven central screens) among the total 17 low-speed streaks. To excite varicose modes, a hole of 3 mm in diameter is drilled just behind the midspan of each screen, 8:5 mm downstream of the screens as illustrated in Fig. 1. Each hole is connected to a loudspeaker by a vinyl hose. To excite sinuous modes, holes of 2 mm in diameter are drilled at both edges of each screen, 13:5 mm downstream of the screens, and are connected to two loudspeakers. The two loudspeakers are driven separately with sine-wave signals that are 180◦ out of phase. In addition, we can select two kinds of excitation for each of varicose and sinuous modes. One is in-phase excitation which can produce fundamental modes having the same phase relation for all the low-speed streaks. The other is anti-phase excitation that is 180◦ out of phase between the neighboring streaks, and it can introduce subharmonic modes. The fundamental modes have the same dominant spanwise wavelength as the streak spacing s , whereas the subharmonic modes have the spanwise wavelength twice the streak spacing 2s . The details of the excited disturbances will be explained later. A constant temperature hot-wire anemometer with a linearizer is used to measure time-mean and 4uctuation velocities in the streamwise direction, U and u. The sensitive length of hot-wire sensor, a tungsten wire of 5 m in diameter, is 1 mm. The hot-wire probe can be traversed in the x, y and z directions. The y-distributions of U and u (the r.m.s. value of u-4uctuation) are recorded in an XY plotter during each y-traverse. The hot-wire data are also stored in a personal computer.
3. Velocity eld downstream of the screens First let us explain the low-speed streaks arti3cially generated by the periodically mounted screens. Fig. 2(a) illustrates the y-distributions of time-mean velocity U at x − x0 = 50 mm for the low-speed streaks generated by the screens of (s ; hs ; ls ) = (12; 2:5; 6 mm). The y-distribution at midspan of each screen is well approximated by a tanh(y) pro3le, whereas the y-distribution between the neighboring screens is almost the same as the Blasius pro3le. Fig. 2(b) illustrates the z-distributions of U measured at several heights, y =1; 2; 3; 4 and 5 mm, at the same x-location. A velocity defect
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Fig. 2. The y- and z-distributions of mean velocity U at x − x0 = 50 mm. (s ; hs ; ls ) = (12; 2:5; 6 mm). (a) y-distributions at z = 6 mm ( ) and z = 0 (), (b) z-distributions at y = 5; 4; 3; 2 and 1 mm.
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due to the drag of the screens is observed markedly below y = 4 mm, and the spanwise velocity distributions across each low-speed streak are not unlike that of a usual wake. The lateral scale of each low-speed streak which is de3ned as the half-value width used for usual wakes is almost the same as (or only slightly smaller than) the width of each screen (ls = 6 mm). We also notice that the velocity excess induced at the side edges of the screen by the longitudinal (stationary horseshoe) vortices is small. This indicates that the eKect of the stationary horseshoe vortex on the velocity 3eld is really weak. Fig. 3 illustrates the streamwise development of low-speed streaks in terms of an iso-velocity contours in the x–z plane at y = 3 mm. The low-speed streaks extend downstream beyond x − x0 = 200 mm (200 mm downstream from the screens) though the velocity defect across each low-speed streak 3lls up gradually with x owing to the momentum transfer by viscous stresses in the lateral and normal-to-wall directions. The slow diKusion of the streak shear layers is due to the fact that the low-speed streaks remain laminar far downstream. The r.m.s. value of u-4uctuations is less than 1% of U∞ even at x − x0 = 200 mm, as shown in Fig. 4 which illustrates the y–z
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Fig. 3. Low-speed streaks downstream of the screens with (s ; hs ; ls )=(12; 2:5; 6 mm) represented by iso-velocity contours in x–z plane at y = 3 mm. Contour levels; U=U∞ = 0:3–0.7.
Fig. 4. The y- and z-distributions of r.m.s. value of u-4uctuations at x − x0 = 200 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm). Contour levels; u1 =U∞ = 0:001–0.005.
distributions of the r.m.s. Value u at x − x0 = 200 mm. The distribution of u takes maxima on the left and right vertical (@U=@z) shear layers and almost vanishes at the midspan z = 0, which indicates the feature of sinuous instability modes. The distribution of sinuous mode will be presented later. Similar periodic low-speed streaks are realized for the screen arrangement with larger spacing s . Fig. 5 compares in three cases of low-speed streaks with s = 12; 15 and 18 mm by iso-velocity contours in the y–z cross-section at x − x0 = 60 mm. The velocity 3eld developing downstream of each screen is almost the same in all the cases. We also con3rmed that the y-distributions at midspan of each screen in these three cases are almost completely the same.
4. Excitation of instability modes The in4ectional velocity pro3les across each low-speed streak are unstable to varicose and sinuous modes. Under background freestream turbulence, this instability nature may amplify ‘natural’ disturbances. In the present low-speed streaks, however, the 4ow remains laminar completely in the
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Fig. 5. Iso-velocity contours in y–z plane at x − x0 = 60 mm. Contour levels; U=U∞ = 0:1–0.9. (a) s = 12 mm, (b) s = 15 mm, (c) s = 18 mm. (hs ; ls ) = (2:5; 6 mm).
whole observation region up to x − x0 = 200 mm owing to low background turbulence less than is only 0.1% of U∞ . As already shown in Fig. 4, the maximum r.m.s. value of u-4uctuations um about 1% of U∞ even at x − x0 = 200 mm. This fact enables us to examine the streak instability by introducing arti3cial disturbances. Unlike the case of a single low-speed streak, interactions between disturbances developing along each low-speed streak might have a signi3cant eKect on the instability in the periodic low-speed streaks. So, we introduced arti3cial disturbances in two diKerent ways to excite fundamental and subharmonic modes for each of the varicose and sinuous instabilities. For the fundamental varicose modes, sinusoidal disturbances given to all the low-speed streaks are in phase, and therefore the excited varicose modes have the spanwise periodicity of the same dominant wavelength s as the low-speed streaks. For the subharmonic varicose modes, on the other hand, the disturbances are 180◦ out of phase between the neighboring streaks and they have the periodicity of twice the streak spacing 2s . These are also the cases for excitation of sinuous modes. The in-phase forcing to all the low-speed streaks excites fundamental sinuous modes having the dominant wavelength s , and the anti-phase forcing to the neighboring low-speed streaks excites subharmonic sinuous modes having the dominant wavelength 2s . Figs. 6(a) and (b), respectively, illustrate the fundamental and subharmonic varicose modes excited in the streaks with the spacing s = 12 mm in terms of instantaneous streamwise velocity 4uctuation u1 (the forcing frequency component of u-4uctuation) in the x–z plane at y=3 mm. Here, the forcing frequency is 90 Hz, and these instantaneous velocity 4uctuation 3elds were obtained by means of phase averaging technique. The disturbances along all the low-speed streaks are in-phase for the fundamental mode, as seen in Fig. 6(a). On the other hand, subharmonic varicose modes have the dominant wavelength of 2s , and therefore we can see a staggered pattern of iso-velocity contours in
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Fig. 6. Instantaneous u-4uctuation 3eld of varicose mode (f = 90 Hz) in the x–z plane at y = 3 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm), represented by iso-velocity lines of u1 =U∞ = −0:01–0.01. (a) Fundamental mode, (b) subharmonic mode.
Fig. 7. Amplitude distributions of varicose modes (f = 90 Hz) in the y–z plane at x − x0 = 60 mm for (s ; hs ; ls ) = (12; 2:5; 6 mm). (a) Fundamental mode, represented by iso-intensity lines of u1 =U∞ = 0:0005–0.0045, (b) subharmonic mode, represented by iso-intensity lines of u1 =U∞ = 0:0006–0.0054.
Fig. 6(b). Note that the staggered pattern in Fig. 6(b) is not unlike that caused by a pair of oblique Tollmien–Schlichting waves which might be expected to appear if the low-speed streaks were so weak that the 4ow was considered to be Blasius 4ow. Figs. 7(a) and (b) illustrate the amplitude distributions of the fundamental and subharmonic varicose modes, respectively, in terms of u1 (the r.m.s. value of u1 ) in the y–z cross-section at x − x0 = 60 mm. Both of the amplitude distributions
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Fig. 8. Instantaneous u-4uctuation 3eld of sinuous mode (f = 60 Hz) in the x–z plane at y = 2:5 mm for (s ; hs ; ls ) = (18; 2:5; 6 mm). (a) Fundamental mode, represented by iso-velocity lines of u1 =U∞ = −0:003–0.003, (b) subharmonic mode, represented by iso-velocity lines of u1 =U∞ = −0:01–0.01.
are not so diKerent from each other except in the high-speed region between the neighboring two screens: The amplitude u1 becomes zero at z = ±6 mm for the subharmonic mode because of the 180◦ phase jump between the neighboring low-speed streaks, while it does not for the fundamental mode. Similarly, Figs. 8(a) and (b), respectively, show the fundamental and subharmonic sinuous modes excited in the streaks with s = 18 mm in terms of instantaneous velocity 4uctuations of forcing frequency (60 Hz) component u1 in the x–z plane at y = 2:5 mm. Superposition of u1 on the base streak 4ow U leads to the wavy structure of the low-speed streaks (or meandering motion of the streaks) in both the cases. Figs. 9(a) and (b) illustrate, respectively, the corresponding amplitude distributions u1 (y; z) of the fundamental and subharmonic sinuous modes at x−x0 =60 mm, showing that both the amplitude distributions across each low-speed streak are not so largely diKerent from each other. It will be seen later that such a slight diKerence in the structure causes a signi3cant diKerence in the disturbance growth. In the following, the development of these sinuous and varicose modes is examined in detail. 5. Development of sinuous modes Fig. 10(a) shows the streamwise development of subharmonic disturbances in the low-speed streaks with the smallest spacing s = 12 mm by plotting u1 m , the maximum amplitude of u1 against x. The forcing frequency is varied from 30 to 90 Hz. The disturbances can grow except 90 Hz. In particular,
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Fig. 9. Amplitude distributions of sinuous modes (f = 60 Hz) in the y–z plane at x − x0 = 60 mm. (s ; hs ; ls ) = (18; 2:5; 6 mm). (a) Fundamental mode, represented by iso-velocity lines of u1 =U∞ = 0:0002–0.0014, (b) subharmonic mode, represented by iso-velocity lines of u1 =U∞ = 0:0003–0.0027.
Fig. 10. Development of subharmonic sinuous modes. (a) s = 12 mm, (b) s = 18 mm. (hs ; ls ) = (2:5; 6 mm): ; 30; ; 45; ; 60; ; 70; O; 80; 5; 90 Hz.
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Fig. 11. Comparison of development of subharmonic sinuous modes excited at f = 60 Hz. (hs ; ls ) = (2:5; 6 mm). ; s = 12 mm; ; s = 15 mm; , s = 18 mm; , single low-speed streak.
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the disturbances of 45 and 60 Hz are ampli3ed about 20 times over the region from x − x0 = 30– 200 mm, and further continue to grow beyond x−x0 =200 mm. We also notice that the growth rate is decreasing gradually. This is because the low-speed streaks become weak in the downstream direction due to viscous diKusion of the associated vertical and horizontal shear layers. Fig. 10(b) illustrates the corresponding result for the low-speed streaks with the spacing s = 18 mm. The frequency of the most ampli3ed disturbance is the same between both the streaks of s = 12 and 18 mm, and the growth rate for s =18 mm is only slightly larger than that for s =12 mm. To show this feature more directly, Fig. 11 compares the streamwise development of the most ampli3ed disturbance (60 Hz) for the three cases of s = 12; 15 and 18 mm. The 3gure also includes the corresponding result for the single low-speed streak generated by using the same-sized screen (hs = 2:5 mm, ls = 6 mm), for comparison. It can be seen that the growth rate soon tends to the value for the single streak with increasing the streak spacing s . The growth rate for s = 18 mm already coincides with that for the single streak. Thus, the constraint of the periodic boundary condition little aKects on the streamwise growth of subharmonic modes. On the other hand, Fig. 12(a) illustrates the development of fundamental sinuous modes for the streak spacing s = 18 mm. The frequency of the most ampli3ed disturbance is around 60 Hz, not diKerent from that of the subharmonic mode. However, the maximum growth rate, estimated by d[ln u1 m ]=d x over x −x0 =50–100 mm, is only one-third that of the subharmonic mode. Furthermore, when the streak spacing s is decreased to 15 mm which gives s =ls = 2:5, the fundamental sinuous modes cease to grow. Fig. 12(b) illustrates the streamwise development of fundamental modes excited at various frequencies for s = 15 mm, showing that the disturbance development is nearly neutral. We also examined the development of fundamental modes in the case of s = 12 mm and con3rmed that there did not exist any growing fundamental disturbances for s = 12 mm. The large diKerence in the disturbance growth between the fundamental and subharmonic modes is attributed to slight diKerences in the disturbance structure. Comparing the u1 -3elds shown in Figs. 9(a) and (b), we can see that the fundamental mode interacts more strongly with disturbances along the neighboring low-speed streaks. As for the subharmonic mode, on the other hand, disturbance
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Fig. 12. Development of fundamental sinuous modes. (a) s = 18 mm, (b) s = 15 mm. (hs ; ls ) = (2:5; 6 mm). ; 45; ; 60; ; 70; O; 80; H; 90 Hz.
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along each low-speed streak seems to behave almost independently, which supports that the growth rate of the subharmonic mode is almost the same as that for the single low-speed streak. Since the sinuous instability is governed by the wake-type velocity distribution across each low-speed streak U (z), the growth rate might depend on the streak height relative to the streak width: The streak height can be de3ned as the height of the in4ection point of the velocity pro3le U (y). In this concern, Schoppa and Hussain (2002) analyzed the linear stability of the near-wall streaks deduced from DNS of wall turbulence and showed that the sinuous instability to fundamental modes could occur only for strong low-speed streaks with wall inclination angle of the streak vortex line above 50◦ . The large inclination angle corresponds to suSciently lifted-up low-speed streaks, so it is reasonable that the critical condition is related to the inclination angle for the streaks with similar geometry. In the present low-speed streaks generated by rectangular screens with the height-to-width ratio hs =ls = 0:42, the inclination angle is about 45◦ or less around x − x0 = 60 mm. If hs =ls is increased, the present low-speed steaks might turn to be unstable even for s =ls = 2:5 or less, though the streak geometry is not the same as that analyzed by Schoppa and Hussain (2002).
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Fig. 13. Iso-velocity lines in y–z plane at x − x0 = 60 mm. (s ; hs ; ls ) = (15; 3; 6 mm). Contour levels; U=U = 0:1–0.9.
In order to con3rm this, we examined the characteristics of sinuous instability by using 1.2 times taller screens with the screen width unchanged, i.e., (hs ; ls ) = (3 mm; 6 mm), which gives hs =ls = 0:5. Fig. 13 shows the velocity distribution in the y–z cross-section at x − x0 = 60 mm for the streaks generated by the screens of hs =ls = 0:5, with spacing s = 15 mm (s =ls = 2:5). Comparing with Fig. 5, it can be understood that the low-speed streaks extend to higher y-position. Therefore it is expected that the low-speed streaks becomes more unstable. The fundamental and subharmonic sinuous modes are excited in this case, and their streamwise development is shown in Figs. 14(a) and (b), respectively. As expected, the subharmonic mode rapidly grows with larger growth rate in this case, compared with the case of hs =ls = 0:42. The growth rate (d[ln u1 m ]=d x) of the most ampli3ed disturbance (60–70 Hz) estimated from the data over x − x0 = 50–100 mm is 0:038 [mm−1 ] for hs =ls = 0:50, while the corresponding growth rate is 0.023 [mm−1 ] for hs =ls = 0:42. The increase in hs =ls also operates to destabilize the low-speed streaks to fundamental modes, as seen in Fig. 14(a), which shows that disturbances of 60 and 70 Hz grow. However, the growth rates are very small, of the order of 0:01 [mm−1 ] and therefore the increase in hs =ls reduces only slightly the critical spacing s =ls for the growth of fundamental mode. Indeed, we con3rmed that the smallest streak spacing of s = 12 mm (s =ls = 2) suppressed the growth of fundamental sinuous modes completely in the case of hs =ls = 0:5, too. Thus, we may say that the critical streak spacing for the growth of fundamental sinuous mode is 2.5 or slightly less in terms of s =ls .
6. Development of varicose modes Figs. 15(a) and (b) illustrate the streamwise development of fundamental and subharmonic varicose modes in the case of s =12 mm (s =ls =2). Both the modes rapidly grow up to x −x0 =70 mm even for this smallest spacing s =ls = 2. The frequency of the most ampli3ed disturbance is at and around 90 Hz in both the cases, which is higher than the frequency of the most ampli3ed sinuous mode. The growth of varicose modes is essentially dominated by the Kelvin–Helmholts-type instability of the in4ectional velocity pro3le U (y), but their development is strongly aKected by the ratio of the streak width to the shear layer thickness. The disturbances undergo large ampli3cation for x − x0 ¡ 60 mm where the shear layer is thin compared with the streak width. As the streak shear layer is diKused by viscous diKusion, the growth rate is gradually reduced with increasing x and 3nally the disturbances cease to grow beyond x − x0 = 80 mm both for the fundamental and subharmonic modes.
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Fig. 14. Development of (a) fundamental and (b) subharmonic sinuous modes. (s ; hs ; ls ) = (15; 3; 6 mm). ; 45; ; 60; ; 70; O; 80; H; 90 Hz.
•
◦; 30;
In order to compare the ampli3cation of these fundamental and subharmonic varicose modes quantitatively, the N -factor de3ned as N = ln[u1 m (x)=u1 m (x1 )] is calculated for each frequency f. Here, x1 is selected to be the location 30 mm downstream of the screens, i.e., x1 − x0 = 30 mm, where the exponential growth has already started. The maximum value of N , which is denoted by Nmax , is obtained at an x-station where the amplitude u1 m attains a maximum for each frequency. Fig. 16 plots Nmax against the forcing frequency f. The 3gure includes the result of the single low-speed streak generated by the same-sized screen, i.e., ls = 6 mm and hs = 2:5 mm. Comparing the Nmax in the three cases, we see that both the fundamental and subharmonic modes are markedly suppressed due to the constraint of the spanwise periodicity imposed on the disturbances for all the frequencies examined. On the most ampli3ed disturbance, the values of Nmax for the fundamental and subharmonic modes are respectively about 50% and 65% smaller than that for the single streak. It is also worth noting that the reduction in the growth rate is more pronounced for higher-frequency disturbances (or smaller-scale disturbances). These results indicate that the three-dimensional nature of the instability is not neglected even though the growth of varicose modes is essentially governed by Kelvin–Helmholts instability of horizontal (@U=@y) shear layer.
Y. Konishi, M. Asai / Fluid Dynamics Research 34 (2004) 299 – 315
Fig. 15. Development of (a) fundamental and (b) subharmonic varicose modes. (s ; hs ; ls ) = (12; 2:5; 6 mm). ; 60; ; 75; H; 90; 4; 110; O; 130 Hz.
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Fig. 16. Nmax versus frequency f for varicose modes. (s ; hs ; ls )=(12; 2:5; 6 mm). mode; , single low-speed streak.
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; 45;
◦, fundamental mode; , subharmonic
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7. Conclusions The instability of spanwise-periodic low-speed streaks was studied experimentally. The structure of the low-speed streaks was well controlled by the lateral spacing (s ) of small pieces of screens as well as their lateral and vertical dimensions (ls and hs ). The low-speed streaks remained laminar far downstream in all the cases examined, which enabled us to examine the linear instability to both sinuous and varicose modes by introducing arti3cial disturbances. The fundamental and subharmonic disturbances which had dominant spanwise wavelenght of s and 2s , respectively, were also excited successfully for each of the sinuous and varicose instabilities. The main results are summarized below. On the sinuous instability, the subharmonic mode can grow with almost the same growth rate as that for the single low-speed streak. When the streak spacing s is three times larger than the streak width ls , the maximum growth rate is almost the same as that for the single low-speed streak. Even for s =ls = 2:0, the growth rate is only slightly smaller than that for the single streak. However, this is not the case for the fundamental sinuous modes. The constraint due to the spanwise periodicity inhibits the ampli3cation of fundamental sinuous modes with dominant wavelength of s when the s =ls is less than a critical value, though the critical streak spacing is weakly dependent on the height-to-width ratio hs =ls . The critical value of s =ls for the growth of fundamental sinuous modes is 2.5 for hs =ls = 0:42 and slightly less for hs =ls = 0:5. Even for larger streak spacing s =ls = 3 with hs =ls = 0:42, the spatial growth rate of the most ampli3ed fundamental mode is only about one-third that of the subharmonic mode. On the varicose instability, both the fundamental and subharmonic modes are less ampli3ed, compared with the case of single low-speed streak. The reduction in the ampli3cation is more pronounced for higher-frequency disturbances. Besides, the fundamental varicose modes are more largely aKected by the periodic constraint than the subharmonic varicose modes. Acknowledgements This work was in part supported by the Grant-in-Aid for Scienti3c Research C (No. 13650963) from the Japan Society for the Promotion of Science and the Grant-in-Aid for Special Scienti3c Research (No. 12125203) from the Ministry of Education, Sports, Culture, Science and Technology, Japan. References Asai, M., Minagawa, M., Nishioka, M., 2000. Instability and breakdown of the three-dimensional high-shear layer associated with a near-wall low-speed streak. In: Fasel, H., Saric, W. (Eds.), Laminar-Turbulent Transition. Springer, Berlin, pp. 269–274. Asai, M., Minagawa, M., Nishioka, M., 2002. The instability and breakdown of a near-wall low-speed streak. J. Fluid Mech. 455, 289–314. Brandt, L., Henningson, D.S., 2002. Transition of streamwise streaks in zero-pressure-gradient boundary layers. J. Fluid Mech. 472, 229–261. Elofsson, P.A., Kawakami, M., Alfredsson, P.H., 1999. Experiments on the stability of streamwise streaks in plane Poiseuille 4ow. Phys. Fluids 11, 915–930.
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