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**Sinusoidal and varicose instabilities of streaks in a boundary layer**

Multimedia files – 8/13 Sinusoidal and varicose instabilities of streaks in a boundary layer Contents: 1. Instabilities of streaky structures 2. Test model 3. Velocity field downstream of the roughness element 4. Antisymmetric sinusoidal instability mode, utot 5. Antisymmetric sinusoidal instability mode, uave 6. Symmetric varicose instability mode, utot 7. Symmetric varicose instability mode, uave 8. Related publications

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**1. Instabilities of streaky structures (I)**

Laminar-turbulent transition in boundary layers is often associated with steady and unsteady basic-flow streamwise perturbations including the streaky structures. Transverse modulation of the wall-bounded flows by the streaks and the inflections induced by them in the instantaneous velocity profiles both in the wall-normal and transverse directions generate conditions for the origination and development of secondary oscillations. They are responsible for the varicose and sinusoidal modes of the secondary disturbances, respectively. Instantaneous velocities in x-z plane: sinous (odd) mode (top) and varicose (even) mode (bottom) (Li & Malik 1995) (see page of notes)

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**1. Instabilities of streaky structures (II)**

Numerical simulation of the sinusoidal instability of streaky structures in a turbulent boundary layer; x-z (top) and x-y (bottom) planes (Brandt 2002) The choice of the instability mode excited first and growing most rapidly depends on the initial conditions, in particular, on a distance between the streamwise disturbances. In the following we deal with the varicose and sinusoidal instabilities of the streamwise streaks in the Blasius boundary layer, most of all, with the coherent structures arising due to the streak instabilities. Numerical simulation of the varicose instability of streaky structure (Skote et al. 2002) (see page of notes)

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2. Test model Excitation of a varicose instability mode Excitation of a streaky structure Excitation of a sinusoidal instability mode Wind-tunnel results (Chernoray et al. 2006) were obtained on a flat plate at controlled excitation of both primary boundary-layer disturbance (stationary streak) and its secondary oscillations. The first of them was generated behind circular roughness element on the plate; the nonstationary perturbations were induced by periodic air suction and blowing through small holes in the model surface as is sketched in the figure. Hot-wire measurements of the streamwise velocity component were performed in an automated mode using a traversing system moving the probe in a 3D space by a specially developed software involving LabVIEW. Approximately 104 spatial points were mapped in each experimental run. The hot-wire data were processed using the MatLab software package. The measured voltage traces were converted into the velocity traces and then phase averaged over 100 disturbance periods to form a matrix uave(x, y, z, t). Flow velocity is decomposed into a mean U and fluctuation u′ components such that uave = U + u′. Subtraction of the base undisturbed flow UB gives the total disturbance velocity utot = uave - UB. Flat-plate model designed for examination of the streak instabilities (see page of notes)

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**3. Velocity field downstream of the roughness element**

(b) Perturbed flow behind the roughness element at x – x0 = 30 mm for the sinusoidal (a) and varicose (b) instabilities. Filled contours show rms amplitude of the secondary high-frequency disturbances u' as m/s. Lines depict contours of mean velocity at levels of 15, 30, 45, 60 and 75% of U The contour diagrams of the streamwise velocity perturbations provide an idea about the structure of the secondary high-frequency disturbances of the sinusoidal and varicose types behind the roughness element. The distribution of the sinusoidal perturbations is symmetric with respect to z = 0, while its phase is subject to a 180-degree jump across the symmetry axis; thus, the amplitude of this mode vanishes at z = 0. Vice versa, the distribution of the varicose mode displays the maximum amplitude at z = 0 and the absence of the transverse jump in phase. The oscillations excited can be identified as the inherent modes of a three-dimensional shear layer related to the streaky structure. Despite a clearly nonlinear character of the excited waves even near their source, the present results correlate very well with similar data of Asai et al. (2002). The propagation velocities of the symmetric and antisymmetric modes are close to the local flow velocities at the profile inflection points along the z and y axis, respectively, where the gradients U/z and U/y attain the highest values. More details about the streak instabilities are available in the following illustrations. (see page of notes)

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**4. Antisymmetric sinusoidal instability mode, utot (I)**

The sinusoidal destruction of the streaky structure is shown by the isosurface diagrams of total instantaneous flow perturbation utot. A very typical for the sinusoidal mode streak meandering during the initial stage of breakdown is seen. Further downstream the coherent structures which resemble the Λ-vortices are generated. The transverse smearing of the disturbed flow region in the downstream direction which is associated with the multiplication of the streaky structures is observed clearly. Streak breakdown through the sinusoidal instability. Isosurfaces of total instantaneous flow perturbation utot (yellow – positive, blue – negative) at different levels: ± 10.3 %, ± 6.4 %, ± 2.6 %, and ± 0.65 % of U, from left to right (see page of notes)

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**4. Antisymmetric sinusoidal instability mode, utot (II)**

Click to play A video clip (hot-wire “visualization”) by Chernoray V.G., Kozlov V.V., Löfdahl L., Litvinenko Yu.A., Grek G.R., Chun H. (2004)

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**5. Antisymmetric sinusoidal instability mode, uave (I)**

The development of the secondary high-frequency disturbance demonstrates its transverse fragmentation into a number of closed regions of isolines near the wall. The streak oscillations (meandering) lead to the generation of a chain of streamwise vortices entrained downstream with their transverse multiplication. At the initial stage of the disturbance evolution one can observe a pair of streamwise velocity deformations of alternating sign which are transformed into the Λ-shaped structures. In a flow pattern with the levels of disturbance amplitude of ± 6.4 % U (left), one can mainly observe quasi-streamwise structures, whereas in the case of the spatial pattern with the disturbance amplitudes of ± 0.65 % U (right), one can see typical coherent structures similar to the Λ-vortices. Thus, the detailed hot-wire measurements of the nonlinear stage of the sinusoidal instability show that the secondary high-frequency destruction of the streaky structure is associated with the formation of Λ-structures whose downstream evolution leads to the flow turbulization. Spatial patterns of the development of high-frequency disturbance of the sinusoidal instability: contours of rms velocity amplitude at x – x0 =114 mm (top); isosurfaces of time-periodic velocity u': ± 6.4 %, 2.6 %, 1.3 %, and 0.65 % of U, from left to right (bottom) (see page of notes)

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**5. Antisymmetric sinusoidal instability mode, uave (II)**

Click to play A video clip (hot-wire “visualization”) by Chernoray V.G., Kozlov V.V., Löfdahl L., Litvinenko Yu.A., Grek G.R., Chun H. (2004)

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**6. Symmetric varicose instability mode, utot (I)**

For the varicose mode, the isosurface diagrams of the total instantaneous flow perturbation also demonstrate the transverse smearing of the disturbed region in the downstream direction, which is related to the multiplication of streaky structures. The spatial pattern of the disturbance evolution shows that the streamwise modulation in the form of streak bunching is observed in the upstream sections, which is typical for the development of the varicose instability. Further downstream the perturbed flow is transformed into typical coherent structures resembling Λ-vortices similarly to that at the sinusoidal streak destruction. However, in contrast to the previous case, three rows of the Λ-structures can be seen which are symmetric on the sides. Streak breakdown through the varicose instability. Isosurfaces of total instantaneous flow perturbation utot (yellow – positive, blue – negative) at different levels: ± 10.3 %, 6.4 %, 2.6 %, and 0.65 % of U, from left to right (see page of notes)

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**6. Symmetric varicose instability mode, utot (II)**

Click to play A video clip (hot-wire “visualization”) by Chernoray V.G., Kozlov V.V., Löfdahl L., Litvinenko Yu.A., Grek G.R., Chun H. (2004)

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**7. Symmetric varicose instability mode, uave (I)**

The secondary high-frequency disturbance generated on the streaky structure reveals the symmetric Λ-shapes located at the centerline which are formed from a sequence of the periodical structures. The process of development of the secondary disturbance is particularly well seen in the flow patterns at different amplitude levels. The diagrams demonstrate transverse fragmentation of the initially well-defined structure into a series of elongated regions, as in the case of the sinusoidal instability. The streamwise modulation of the streak leads to the generation of hairpin or a pair of counter-rotating quasi-streamwise vortices. At the transverse boundaries of the disturbed region, the Λ-structures or hairpin vortices appear to be asymmetric; nevertheless, the second counter-rotating ‘leg’ of these coherent structures can be observed clearly as forming. Thus, the transverse multiplication of the longitudinal structures takes place. It can be deduced that the low-velocity fluid is moving upward there, and each ‘leg’ of the hairpin vortex evolves into a pair of counter-rotating streamwise vortices on each period of the secondary disturbance. At the amplitude levels of ± 2.6 % of U, localized structures modulating the low-speed streak are visible which are multiplied in transverse direction with emergence of Λ-vortices. The flow pattern at the level of the disturbance amplitude of ± 0.4 % U shows the origination of Λ-vortices at a significantly earlier stage. Spatial patterns of the development of high-frequency disturbance of the varicose instability: contours of rms velocity amplitude at x – x0 =114 mm (top); isosurfaces of time-periodic velocity : ± 2.6%, 1.3%, 0.65% and 0.4% of U, from left to right (bottom) (see page of notes)

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**7. Symmetric varicose instability mode, uave (II)**

Click to play Finally, recall that the nonlinear stage of the ‘classical’ scenario of the laminar-turbulent transition is associated with a three-dimensional distortion of the two-dimensional Tollmien-Schlichting waves and the formation of three-dimensional coherent structures such as Λ-vortices. The above data show that there are other routes of the Λ-structures origination in the near-wall shear flows, in particular due to the secondary high-frequency sinusoidal and varicose streaks instabilities. A video clip (hot-wire “visualization”) by Chernoray V.G., Kozlov V.V., Löfdahl L., Litvinenko Yu.A., Grek G.R., Chun H. (2004) (see page of notes)

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**8. Related publications (I)**

Adrian R.J., Meinhart C.D., Tomkins C.D. (2000) Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech., 422, 1-23. Asai M., Minagawa M., Nishioka M. (2002) The stability and breakdown of near-wall low-speed streak. J. Fluid Mech., 455, Bippes H. (1972) Experimentelle Untersuchung des laminar-turbulenten Umschlags an einer parallel angestroemten konkaven Wand. Sitzungsberichte der Heidelberger Akademie der Wissenschaften Mathematisch-naturwissenschaftliche Klasse, Jahrgang, 3 Abhandlung, (also NASA-TM-72243, March 1978). Brandt L., Heningsson D.S. (2002) Transition of streamwise streaks in zero-pressure-gradient boundary layers. J. Fluid Mech., 472, Chernoray, V.G., Kozlov, V.V., Löfdahl, L., Chun, H.H. (2006) Visualization of sinusoidal and varicose instabilities of streaks in a boundary layer . J. Visualization, 9(4), Litvinenko Yu.A., Chernorai V.G., Kozlov V.V., Loefdahl L., Grek G.R., Chun H.H. (2004) Nonlinear sinusoidal and varicose instability in the boundary layer (Review). Thermophysics and Aeromechanics, 11(3), Litvinenko Yu.A., Chernorai V.G., Kozlov V.V., Loefdahl L., Grek G.R., Chun H.H. (2005) Nonlinear sinusoidal and varicose instability in the boundary layer. Doklady Physics, 50(3), 147–150. Haidary H.A., Smith C.R. (1994) The generation and regeneration of single hairpin vortices. J. Fluid Mech., 227, Hamilton H., Kim J., Waleffe F. (1995) Regeneration of near-wall turbulence structures. J. Fluid Mech., 287, Ito A. (1985) Breakdown structure of longitudinal vortices along a concave wall. J. Japan Soc. Aero. Space Sci., 33, Kawahara G., Jimenez J., Uhlmann M., Pinelli A. (1998) The instability of streaks in near-wall turbulence. Center for Turbulence Research, Annual Research Briefs,

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**8. Related publications (II)**

Konishi Y., Asai M. (2004) Experimental investigation of the instability of spanwise-periodic low-speed streaks in a laminar boundary layer. Japan Fluid Mech. J., 021257, Panton R.L. (2001) Overview of the self-sustaining mechanisms of wall turbulence. Progr. Aerosp. Sci., 37, Robinson S.K. (1991) The kinematics of turbulent boundary layer structure. NASA TM Schoppa W., Hussain F. (1997) Genesis and dynamics of coherent structures in near-wall turbulence: A new look. Self-sustaining Mechanisms of Wall Turbulence, R.L. Panton (ed.), Computational mechanics, Southampton. Skote M., Haritonidis J.H., Henningson D.S. (2002) Varicose instabilities in turbulent boundary layers. Phys. Fluids, 4(7),

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