theory of boundary layer instability: particle or wave?

Symposium "One Hundred Years of Boundary Layer Research" August 12 - 14, 2004 DLR Göttingen, Germany

THEORY OF BOUNDARY LAYER INSTABILITY: PARTICLE OR WAVE? Ka-Kheng Tan Department of Chemical and Environmental Engineering, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia [email protected] or [email protected]. Fax: 603 8656 7200 Abstract-- No stability analyses of a growing boundary layer provides a unique solution to the position of the onset of instability because the point of instability is not known a priori, and the underlying equations may be normalized by any thickness of the boundary layer. Tollmien [1] has admitted in the first theoretical study of boundary-layer instability that his choice of and the definition of the displacement thickness δ1 is fundamentally unscientific and problematic. Therefore it is not surprising that the value of the conventional theoretical Reynolds number

(Re ), based δ1

as it is on an

arbitrarily-defined displacement thickness, has failed to give a theoretical value of longitudinal Reynolds number (Rex), that agrees with experimental observations. It is found in this study that the onset of instability can be predicted from a differential Reynolds number

Re y , defined as y 2 (∂u ∂y ) ν , whose maximum value for the case

of the Blasius velocity profile is located at maximum Reynolds number at this

y c = 2.95 νx U ∞ critical

depth

is

or = 0.6δ . The found to be

Re yc = 1.454U∞ νx U∞ ν = 1.454 Re x c

, and shows a theoretical critical value of 681 and a corresponding critical value of Rex of 219 000. This has been verified by a plot of 210 000 and

cf

against Rex that shows a departure from Blasius flow at a critical Rex =

c f = 0.0029

for several experiments and a numerical study. Thus the

onset of instability is marked by the deviation from Newton’s law of friction, which is

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followed by the onset of convection of momentum that characterizes the transition to turbulence. This particle approach to boundary layer instability is superior to the socalled wave-amplification theory and its verification experiments of artificial wave amplifications Key words: boundary layer, displacement thickness, Reynolds number and instability.

1. INTRODUCTION Prandtl [2] discovered the boundary layer early last century, although Lord Rayleigh [3] was the first to investigate the stability of laminar flow, albeit both did not provide a criterion of instability. It is wellestablished that all boundary layer flows are inherently unstable if the plates are sufficiently long [4]. However, to date the theory of the boundary-layer instability on a flat plate is not in full agreement with experimental observations, particularly in the position and the origin of the onset of instability in laminar flow, in that the theoretical value of critical Reynolds number Re δ 1 (= U ∞ δ 1 ν ) of 520 from stability analyses [5] and [6] can not provide a correct critical Rex, which also cannot be determined unambiguously in wave-amplification experiments [7] and [8]. The cause of the discrepancy between theory and experiments may be traced to the first theoretical study of [1], who recognized that his approach to determine the interface (“Grenzschichtdicke”) thickness, now known as displacement thickness

δ 1 = 1.72 νx U ∞ , from the basic fundamental science is unscientific and flawed. He has deterministically defined the displacement thickness from a flow balance to delineate the portion of the flow stream in the boundary layer that would have a free stream velocity U∞ , i.e.

U ∞δ 1 = ∫

∞

y =0

(U ∞ − u )dy .

The integral merely represents the imaginary

reduced or displaced flow due to the effect of friction; thus, the displacement thickness is fundamentally irrational and problematic as it is not the point of instability. He also discovered that the value of transition Rex of 300 000 from experiments of Hansen [9] yielded a large Re δ1 of 950, which is more than double his theoretical value of 420. Conversely his theoretical value of 420 would yield a critical Rex of only 59, 600. Later [7] repeated this calculation with the more accurate value of Reδ 1 = 520 and also reached the same conclusion. It is clear that the so-called displacement thickness is merely a mathematical convenience for providing a sensible scaling length for the definition of a Reynolds number in any stability analysis, it later became a convention that defied critical review. Indeed, all stability analyses, be it linear stability analysis (LSA), parabolized stability equations (PSE) or direct numerical simulations (DNS), do not yield a unique solution as the critical point of instability is not known a priori; hence, they may be normalized by any arbitrary length of the boundary layer.

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Tollmien’s [1] failure to obtain a correct critical Rex from his theory is perhaps explainable since the formulation of the Reynolds number Re δ1 hints at a linear velocity gradient whereas the velocity profile is non-linear.

The linear form of Reynolds number Re δ 1 = U ∞ δ 1 ν

implied a linear velocity gradient du dy = U ∞ δ 1 , which is roughly twice that of the Blasius velocity profile at δ 1 = 1.72 νx U ∞ , where du dy ≈ 0.5U∞ δ1 . Consequently the resultant critical Rex derived from

Reδ 1 based on du dy = U ∞ δ 1 with a theoretical critical value of 520 will be substantially less than the expected theoretical value of Rex. Saric [10] was critical of the inadequacy of δ1, however, his attempt to use a

reference thickness δr in defining a Re δ r = Re x is also equally futile, if not adding to more confusion, since it results in an unusually small value of Reδ r = 520 1.72 = 302 . Lim [11] has calculated with a simplified linear velocity profile a value of Reδ 1 of 519, which is close to the commonly accepted value of 520. It is thus possible to define a Reδ 1 based on the linear velocity gradient of du dy ≈ 0.5U∞ δ1 , that is Reδ 1 = 0.5U∞δ1 ν , and arrive at a correct critical Rex of about 216 000, based on Reδ 1 ≅ 400 to account for the leading-edge effect (see later). However, the agreement is fortuitous since δ1 is not the point of instability. This may be solved with the theory of transient instability of Tan and Thorpe [12-16], who had successfully predicted the onset of transient buoyancy, Marangoni and momentum instability induced by unsteady-state diffusions, which are characterized by nonlinear temperature, concentration and velocity profiles respectively. The onsets of these instabilities were easily predicted accurately with their newly defined transient Rayleigh numbers, transient Marangoni and transient Taylor numbers respectively that incorporated local temperature, concentration and velocity gradients. Tan [16] had systematically derived a unified transient instability theory from a comprehensive study of this broad class of instability, it will be applied to the present study of boundary layer instability. Hinze’s [17] detailed review of the mechanism of instability and transition revealed that the experimental velocity profile showed a rounded kink at a point approximately y = 0.6δ , i.e. 3 νx U ∞ in the boundary layer. Schlichting [18] and Kurtz and Crandall [19], Kaplan [20] and [5] found in their calculations and Schubauer and Skramstad [21] in their experiments that the perturbation velocity underwent 180˚ phase shift at y = 3 ν x U∞ in the region of “transition”. Klebanov and Tidstrom [22] and later Kovasznay et al. [23] found that breakdown of local flow occurred at y = 0.6δ . Perhaps the best measurement of

y c = 0.6δ was provided by the experiments of Klingmann et al. [24]

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and Swearingen and Blackwelder [25], when the filamentous convection plumes first appeared. Thus, the point of instability should be characterized by this critical depth and not by the so-called displacement thickness. The mechanism of the onset of instability in flows is generally known to be caused by the local in-equilibrium between the inertial and viscous forces, which can be discerned from the momentum equation for a boundary layer with zero-pressure gradient and du dx = 0 :

(

)

v(du dy ) = ν d 2 u dy 2 . If viscosity is the cause of instability in flows, then the friction drag at the point of onset of instability will rise abruptly. The defining test for the onset of instability is then the determination of the point of deviation from the theoretical friction drag 2 coefficient for Blasius flows given by cf = 1.328Re -1 x . Comprehensive measurements of friction drag of Hislop [26] and Liepmann and Dhawan [27] showed a departure from laminar flow at a critical Rex of about 210 000 and c f = 0.0029 , Figure 1. This has also been observed for a boundary layer on a slightly-curved plate by [25] at critical Rex ~ 210 000 and cf = 0.0029 , which have been successfully simulated by Lee and Liu [28].

0.010

(c )

−1 2

)

Coefficient of friction,

cf

f

= 415 log (Re c f

Hislop (1940) Lipmann & Dhawan (1951) Swearingen and Blackwelder (1987) Lee & Liu (1991) Schoenherr (1932) 3ft roughend L.E. Schoenherr (1932) 3ft rounded L.E.

Turbulent

c f = 1.328 Re −1 2 Blasius Laminar

0.001

1 x 104

105 106 Reynolds Number, Rex

Figure 1. Onset of instability caused by momentum diffusion on a flat plate.

Historically the c f in the transition regime for flat plate has not been measured accurately primarily because most past researchers were not aware of the emergence of the slow and fast streaks at the onset of instability, which will entail the measurement of the average friction

4

107

drag in the spanwise direction. The local values of du dy of slow and fast streaks measured by [25] and calculated by [28] appear to be a first attempt to quantify the effect of friction drag on instability. We have computed the span-wise average c f from their data and accurately determined the onset of instability as shown in Figure 1. All these experiments of low disturbance flows support the theory that the onset of instability occur at a critical value of Rex of 210 000. Experiments of Shoenherr [29] with a flat plate moving in quiescent water also showed that the point of onset of instability is located at Rex ~ 200 000, and roughness significantly reduced it to about 130 000. Surprisingly this simple conclusive verification of the boundary-layer theory has not been recognized, nor was there any serious attempt in addressing [1] failure to determine a correct value of critical Rex. The onset of instability is clearly delineated by the departure from Newton’s [30] law of friction or momentum diffusion, and the ensuing onset of convection of momentum is the well-known transition to turbulence. It is interesting to note that G. I. Taylor had, thirteen years after the first publication of his unprecedented theory and experiments [31] for steady-state Couette flow instability, felt compelled to measure the friction drag between the cylinders as the only direct test for his theory. He [32] accurately detected the onset of instability at a critical cf = 0.0126 with a plot of the friction drag versus Reynolds number (Red) when the inner cylinder is moving. The corresponding critical

(

)

Taylor number Ta = U i d ν Ri can now be calculated as 1650, which is close to his theoretical value of 1706. His experiments confirm that the onset of instability in flows is the cause of increased friction, and not the amplification of waves. Recently [16] had shown that transient Couette instability for an impulsively started cylinder in a large gap, Chen and Christensen [33] and Taneda [34] is analogous to the onset of instability in a deep tank of fluid heated from below, where thin narrow plumes first appeared and later developed into mushroom-shaped convective plumes. They defined a new transient Taylor number 2 5 2 Ta = z (∂u ∂ z) ν Ri that incorporated the local velocity gradient and the penetration depth and determined the maximum transient Taylor 2

3

2

( )

3

ν Ri . They showed that the point of number, Tamax = 1.461U i νt instability for the experiments of Kirchner and Chen [35] occurred at a transient Taylor number of 1100, which is similar in magnitude to that of a critical Rayleigh number for thermal instability in a fluid layer bounded by a free and a solid surface. The experimental critical dimensionless wavenumber of 3.05 also agrees well with the theoretical value of 2.9. These Couette instability studies seem to confirm that the stability analyses of 2D laminar flows can only provide the criterion of the onset of instability. 2

2

5

Experiments of wave amplifications first performed by [21] suffered from a serious drawback, in that they could not detect the very small perturbation, which was replaced by artificial disturbance generated by electromagnetically-vibrated ribbon. Indeed Saric’s [36] comprehensive review had confirmed that natural disturbances in the free stream were too small for detection. Moreover, the artificial wave experiments measured only a value of Re δ1 = 400, which was lower than the theoretical value of 520. The discrepancy has been found to be caused by the leading edge. Therefore, the onset of instability on a flat plate will inevitably occur at a critical value of Re δ1 approximately 400 [37]. Indeed, the purported experiments of flow instability have been reduced to the investigation of the influence of fluid flow on an artificial wave. Strictly conventional perturbation theory only traces the initial growth of the perturbation up to the point of instability, while the wave equations of perturbed variables merely provide a method to determine the minimum value of Reynolds number that locates the position of the point of instability. Fundamentally, the artificial waves that have substituted the undetectable elusive waves of natural flows are not identical to the hypothetical waves of the stability theory. Thus, the onset of instability in natural flows may not necessarily be caused by wave amplifications. All perturbation analyses of thermal instability Lord Rayleigh [38] and Pearson [39] and Taylor instability [31] also employed wave equations with a wave frequency, but they never relied on the measurements of the wave amplifications to verify the theory. Rather, the heat flux Schmidt and Milverton [40] and Silveston [41] and friction drag [32] respectively were measured to detect the onset of instability at a critical Rayleigh number and Taylor number. In summary the absence of a unique solution of the OSE, PSE and DNS implies the impossibility of determining the point of onset of instability unambiguously, both theoretically and experimentally. The c f − Re x plot employing Newton’s law of friction seems to be the only correct and reliable method in determining the position of instability. This paper will propose a new theory of the onset of instability with the definition of a new spatially transient Reynolds number that will determine the point of onset of instability and the maximum transient Reynolds number, from which the theoretical value of Re x c can be determined accurately.

2. SPATIALLY TRANSIENT REYNOLDS NUMBER The flow of fluid on a flat plate experiences progressive retardation due to the viscous shear of the fluid. At a critical distance downstream the momentary imbalance between the local inertial force and the frictional shear may result in local instability. The position of the point of instability may be simply determined from a maximum transient

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Reynolds number. The mathematics for determining the onset of instability is similar to that developed for the onset of instability induced by an impulsively-started rotating cylinder [16]. The rapidly changing velocity gradients (∂u ∂y ) in the wall region is the primary driving force of instability, which is readily seen in the momentum equation for a boundary layer with zero pressure gradient and du dx = 0 :

(

)

v(du dy ) = ν d 2 u dy 2 . The critical point may be determined by defining a new spatially transient Reynolds number in terms of the distance from the plate y and the varying local velocity gradient as:

Re y =

y 2  ∂u    ν  ∂y 

(1)

The local velocity gradient in the laminar boundary layer is given by the Blasius velocity profile [42] and substituting it in Equation U∞ y 2 Re y = f " (η) ν 3 x U∞ (1), becomes (2) U ∞ νx U ∞ 2 = η f " (η)

ν

where η = y U ∞ ν x , f (η ) = ψ / U ∞ xν and u = U ∞ f ' (η ) .

The

values of η and f ′′ are provided by table of Blasius functions [7]. The maximum value of the transient Reynolds number can be found by plotting Rey against η as shown in Figure 2, which shows the maximum Rey at

ηc = 2.95

(3)

that gives the critical depth at y c = 2.95 νx U ∞ . This agrees remarkably well with theoretical calculations and experimental observations discussed previously, that instability originates at the point y c = 3 νx U ∞ = 0.6δ . This is the critical depth required to normalize the stability equations that may yield the point of instability unambiguously.

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1.6 1.4 1.2

Rey/√Rex

1.0 0.8 0.6 0.4 0.2 0.0

0

2

η

4

6

Figure 2. Variation of Reynolds number in the boundary layer

The maximum transient Reynolds number at the critical depth is easily determined by substituting ηc = 2.95 and the value of f " (η ) = 0.1614 at ηc = 2.95 in Equation (2) as: Re yc = 1.454 Re x c

(4)

The formulation of the new spatially transient Reynolds number is reasonable as the fluid flows with a mean velocity of nearly 0.5U∞ through the critical depth y c = 3 νx U ∞ , and thus the Reynolds number is = 0.5U ∞ × 3 νx U∞ ν = 1.5U ∞ ν x U ∞ ν or 1.5 Re x . The variation of Reynolds numbers with the depth of boundary layer is shown in Figure 2, which has a peak value at ηc = 2.95 and drops sharply thereafter. Reynolds numbers after the peak value of the onset of instability have no more meaningful values because the laminar Blasius velocity profile would have broken down and convection commenced. There is no known theoretical value of Reynolds number based on this critical depth yc. However, Hinze [17] has shown that Reynolds number at δ = 5 ν x U∞ can be extrapolated linearly from Reynolds number at the displacement thickness from linear stability analysis, since the Reynolds number is a function of the thickness of the boundary layer. Therefore, the value of critical Reynolds number at the critical depth yc for

δ e = 1.454 νx U∞

may

be

extrapolated

linearly

from

Re δ1 = 0.854 Re x at δ 1 = 1.72 νx U ∞ with the theoretical value of

8

Re δ1 = 400 , and is found to be Re y = 681. The critical Reynolds c number for the horizontal distance x can now be found from

Re xc = (Reδ e /1.454) as 219 000, which is close to the experimental 2

value of 210, 000 as shown in Figure 1. This same critical value of Rex may

also

be

easily

obtained

Re δ 1 = 400

for

from

Re xc = (400 0.854 ) = 219 000 . More accurate measurements of [26], 2

[27], [25] and numerical calculations of [28] showed that the leadingedge effect will lead to a Re x c = 210 000 . The experimental value of critical Re xc = 210 000 from Figure 1 is easily shown by Re y c = 1.454 Re x c , to give the value of maximum transient Reynolds number or Re y of 666, which is close to the linearly c

extrapolated theoretical value of Re y of 681. The value of Re y of 666 c c results in a Reδ of 391, which is close to the experimental value of 1

Ross et al. [42], Strazisar et al. [43] and [5]. The critical distance for the onset of instability can be found from Equation (4) with a critical Reynolds number of 210 000 as x c = 2.10 ×10 5 ν U∞ . For Blasius flows the theoretical critical c f at

Re xc = 210 000 is found to be 0.0029 from c f = 1.328Re−1 2. There thus exists a critical shear that causes the onset of instability, and is given by τ 0 c = 0.0029 0.5 ρU∞2 = 0.00145 ρU∞2 , which provides a very

(

)

simple and quick calculation of the critical shear for a fluid with a known velocity. The critical dimensionless wavenumber based on the ~ critical depth is found to be a c = ac y c = 0.516 , since the one

~ normalized with the displacement thickness is a c = a cδ 1 = 0.301 [5]. The critical horizontal wavelength of the plume is therefore

λc = 2πy c a~c = 12.18 y c = 35.9 vxc U ∞ , which may be expressed in terms

of

the

as λc = 35.9 Re xc

critical 1/ 2

Reynolds

number

of

219

000

ν / U ∞ = 16450ν / U ∞ , which shows that the λc is

dependent only on the free-stream velocity.

3. CONCLUSIONS It is found that all stability analyses that have been normalized by arbitrary length scale cannot provide a unique solution for the position of instability. This is because one does not know a priori nor a posteiori from these analyses the critical depth within the boundary layer at which instability originates. The origin of instability is found to be located at a

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critical depth y c = 3 νx U ∞ or = 0.6δ at a critical horizontal distance from the leading edge x c = 2.19 ×10 5ν /U∞ . The critical depth may be used for the normalization of the stability equations so as to yield a unique solution that may predict the onset of instability unambiguously. This has provided a rational and scientific basis for determining the critical depth of the boundary layer where instability originates. The new differential form of Reynolds number can successfully predict the onset of instability at the critical point by Equation (4), i.e. Re y = 1.454 Re x . The theoretical values of Reynolds number at the c

c

onset of instability are found to be Re yc = 681 and Rexc = 219 000, which are in good agreement with the average experimental values of 670 and 210 000, respectively. The onsets of instability in several studies were found to occur at a critical friction drag coefficient of 0.0029, which agrees very well with the theoretical value of 0.0029. The maximum c f at the point of transition to turbulence is about 0.006, which is about twice the value of critical c f for the onset of instability.

REFERENCES 1. Tollmien, W. “Uber die entstehung der Turrbulenz”, 1. Mitt. Nachr. Fes. Wiss. Gottingen, Math. Phys. Klasse, pp. 21-44, 1929; English translations in NACA TM 609, 1931. 2. Prandtl, L. “Über Flüssigkeitsbewegung bei sehr kleiner Reibung, or Fluid motion with very small friction”, Proc. Third Intern. Math. Congr., Heidelberg, 1904, pp. 484-491, 1904. Reprinted in: NACA TM 452 (1928) 3. Lord Rayleigh. “On the stability or instability of certain fluid motions”, Proc. London Mathematics Society, 11, pp. 57-70, 1880. 4. Drazin, P. G. Personal communications by e-mail 5th July 2000. “So the critical value of the Re gained from the OSE tells more of where the boundary layer becomes unstable, rather than indicating whether it is stable or unstable. In fact all boundary layers on plates with large chords length are unstable.” 5. Jordinson, R. “The flat plate boundary layer. Part 1. Numerical integration of OrrSommerfeld equation”, Jounal of Fluid Mechanics, 43, 801, 1970. 6. Gaster, M. “On the effects of boundary layer growth on flow stability”, Journal of Fluid Mechanics, 43, pp. 813-818, 1974. 7. Schlichting, H. “Boundary-Layer Theory” 1979. (translated by J. Kestin. McGrawHill, 7th ed., chap. V and XVII). Also 8th ed. revised by Gersten, K., 2000.

8. Saric, W. S., White, E. B., & Reed, H. L. “Boundary layer ‘receptivity to free stream disturbances and its role in transitions”, Invited paper at 30th AIAA Fluid Dynamics

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Conference, 28 June – 1st July 1999, Norfolk, VA, American Institute of Aeronautics and Astronautics, 99-3788, 1999. 9. Hansen, M. “Die Geschwindigkeitsverteilung in der Grenzschicht an einer eingetauchten Platte”, ZAMM 8, pp. 185-199, 1928. (NACA TM 585, 1930). 10. Saric, W. S. “Skin fiction drag reduction”, AGARD Report 786, 1992. 11. Lim, C. W. (2003) personal communications. Please refer to: C.W. Lim, The stability of flow over periodically supported plates, M. Eng. thesis, National University of Singapore, November, 1991, and K.S. Yeo and C.W. Lim, The stability of flow over periodically supported plates -- Potential flow, J. Fluid Structure, 8, 331-354, 1994. 12. Tan, K. K. & Thorpe, R. B. “The onset of convection caused by buoyancy during transient heat conduction in deep fluids”, Chemical Engineering Science, 51, 17, 4127, 1996. 13. Tan, K. K. & Thorpe, R. B. “The onset of convection driven by buoyancy caused by transient heat conduction: Part I. Transient Rayleigh numbers”, Chemical Engineering Science, 54, 225, 1999a. 14. Tan, K. K. & Thorpe, R. B. “The onset of convection driven by buoyancy caused by transient heat conduction: Part II: The Sizes of Plumes”, Chemical Engineering Science, 54, 239, 1999b. 15. Tan, K. K. & Thorpe, R. B. “On convection driven by surface tension caused by transient cooling”, Chemical Engineering Science, 54, 775, 1999c. 16. Tan, K. K. & Thorpe, R. B. “Transient Instability Of The Flow Induced By An Impulsively Started Rotating Cylinder”, Chemical Engineering Science, 58, pp. 149156, 2003. 17. Hinze, J. O. (1975). Turbulence, 2nd ed. (McGraw-Hill, New York,). 18. Schlichting, H. “Amplitudenverteilung und Energiebilanz der kleinen storungen bei der Plattenstromung” Nachr. Ges. wiss. Gottingen, Mathematics Physics, Klasse, Fachgruppe I, 1. pp. 47-78, 1935. 19. Kurtz, E. F., & Crandall, S. H. “Computer-aided analysis of hydrodynamic stability”, Journal of Mathematics Physics, 44, 264, 1962. 20. Kaplan, R. E. (1964). “The stability of laminar incompressible boundary layers in the presence of compliant boundaries” (Ph.D. Thesis, Massachusetts Institute of Technology, Aero-Elastic and Structures Research Laboratory, ASRL TR 116-1) 21. Schubauer, G. B., & Skramstad, H. K. “Laminar boundary layer oscillations and stability of laminar flow”, Journal of Aeronautical Science, 14, pp. 69-78, 1947. Also Laminar boundary layer oscillations on a flat plate. NACA Report, 909. 22. Klebanov, P. S., Tidstrom, K. D. & Sargent, L. M. “The three-dimensional nature of boundary layer instability”, Journal of Fluid Mechanics, 40, pp. 39-47, 1962. 23. Kovasznay, L. S. G., Komoda, H. & Vsudeva, B. R. “Detailed flow field in transition”, Proc. of the 1962 Heat transfer and Fluid Mechanics Institute (Stanford University Press, Stanford, California) pp 1-26, 1962.

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24. Klingmann, B. G. B., Boiko, A. V., Westin, K. J. A., Kozlov, V. V. & Alfredsson. “Experiments on the stability of Tollmien-Schlichting waves”, European Journal Mechanics, B/ Fluids, 12, no. 4, pp. 493-514, 1993. 25. Swearingen, J. D. & Blackwelder, R. F. “The growth and breakdown of streamwise vortices in the presence of a wall”, Journal of Fluid Mechanics, 182, pp. 255-290, 1987. 26. Hislop, G. S. (1940). “The transition of a laminar boundary layer in a wind tunnel”, PhD thesis, Department of Engineering, University of Cambridge. 27. Liepmann, H.W. and Dhawan, S. “Direct measurements of skin friction in low-speed and high-speed flow”, Proc. First US Nat. Congr. Appl. Mech. 869, 1951. 28. Lee, K. & Liu, J. T. C. “On the growth of mushroom-like structures in nonlinear spatially developing Goertler vortex flow”, Physics of Fluids A, 4, pp. 95-103, 1992. 29. Schoenherr, K. E. “Resistance of Flat surfaces moving through a fluid”, Trans. Society. Nav. Arch. And Mar. Eng., 40, 279, 1932. ( See Emmons, 1951) 30. Newton, I. (1687). Principia Mathematica, Book II, Section IX: The circular motion of fluids. 31. Taylor, G. I. “Stability of a viscous liquid contained between two rotating cylinders", Phil. Trans. Roy. Society, (London) A223, 289, 1923. 32. Taylor, G. I. “Fluid friction between two rotating cylinders”, Proc. Roy. Soc., A157, 546-64 and 565-78, 1936. 33. Chen, C. F. & Christensen, D. K. “Stability of flow induced by an impulsively started rotating cylinder” Physics of Fluid, 10, 8, 1845, 1967. 34. Taneda, S. “Visual study of unsteady separated flows around bodies”, Prog. Aerospace Science, 17, pp. 287, 1977. Also in An Album of Fluid Motion, (edited by M. van Dyke, The Parabolic Press, 1982). 35. Kirchner, R. P. & Chen, C. F. “Stability of time-dependent rotational Coutte flow. Part 1. Experimental investigation”, Journal of Fluid Mechanics, 40, 39, 1970. 36. Saric, W.S. “Progress in transition modeling”, AGARD Report 793, 1993. 37. Herbert, T. “The dimensional phenomena in the transitional flat-plate boundary layer”, American Institute of Aeronautics and Astronautics, 85-0489, 1985. 38. Lord Rayleigh. “On convective currents in a horizontal layer of fluid when the higher temperature is on the under side” Phil. Magazine. 32, 529, 1916. 39. Pearson, J. R. A. “On convection cells induced by surface tension”, J. Fluid Mech., 4, 489, 1958. 40. Schmidt, R. J., & Milverton, S. W. “On the instability of a fluid when heated from below”, Proc. Ray. Soc., A 152, 586, 1935. 41. Silveston, P. L. “Wärmedurchgang in Waagerechten Flüssigkeitsschichten Forch”, Gebiete Ingenieurwes. 24, pp. 29-32 & pp. 59-69, 1958.

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42. Ross, J. A., Barnes, F. H., Burns, J. G., & Ross, M. A. S. “The flat plate boundary layer. Part 3. Comparison of theory with experiment”, Journal of Fluid Mechanics, 43, 819, 1970. 43. Strazisar, A. J., Prahc, J. M. & Reshotko, E. “Experimental study of the study of heated boundary layers in waters”. FTAS/TR -75 -113, 1976. Case Western University, Department Fluid Thermal Aerospace Science.

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