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Boundary-Layer Stability and Transition W. S. Saric Arizona State University Tempe, USA Abstract Within the last five years, increased emphasis on secondary instability analysis along with the experimental observations of subharmonic instabilities have changed the picture of the transition process for boundary layers in low-disturbance environments. Additional efforts with Navier-Stokes computations have formed an impressive triad of tools that are beginning to unravel the details of the early stages of transition. This paper reviews these recent efforts. Symbols a chordwise complex wavenumber normalized by disturbance amplitude amplitude at R=Ro, usually Branch I pressure coefficient F co/R= 6.28fv/Uo2: dimensionless frequency f dimensional frequency thz] L ~vx*/UO: boundary-Layer reference length. N ln(A/AO): amplification factor R SIR,, = UoL/v boundary-layer Reynolds number Rat initial boundary-layer Reynolds number, usually Branch I Rx UOx*/v: x-Reynolds number or chord Reynolds number U basic-state chordwise velocity normalized by UO Uo freestream velocity, [m/s] v kinematic viscosity Em2/s] cl) 2~fL/Uo: dimensionless circular frequency x* dimensional chordwise coordinate [m] x chordwise coordinate normalized with L 23 y normal-to-the-wall coordinate z spanwise coordinate 1. Introduction The problems of understanding the origins of turbulent flow and transition to turbulent flow are the most important unsolved problems of fluid mechanics and aerodynamics. There is no dearth of applications for information regarding transition location and the details of the subsequent turbulent flow. A few examples can be given here. (1) Nose cone and heat shield requirements on reentry vehicles and the "aerospace airplane" are critical functions of transition altitude. (2) Vehicle dynamics and "observables" are modulated by the occurrence of laminar-turbulent transition. (3) Should transition be delayed with Laminar Flow Control on the wings of large transport aircraft, a 25% savings in fuel will result. (4) Lack of a reliable transition prediction scheme hampers efforts to accurately predict airfoil surface heat transfer and to cool the blades and vanes in gas turbine engines. (5) The performance and detection of submarines and torpedoes are significantly influenced by turbulent boundary-layer flows and efforts directed toward drag reduction require the details of the turbulent processes. (6) Separation and stall on low-Reynolds-number airfoils and turbine blades strongly depend on whether the boundary layer is laminar, transitional, or turbulent. The common thread connecting each of these applications is the fact that they all deal with bounded shear flows (boundary layers) in open systems (with different upstream or initial amplitude conditions). It is well known that the stability, transition, and turbulent characteristics of bounded shear layers are fundamentally different from those of free shear layers (Morkovin, 1969; Tani, 1969; Reshotko, 1976~. Likewise, the stability, transition, and turbulent characteristics of open systems are fundamentally different from those of closed systems (Tatsumi, 1984~. The distinctions are vital. Because of the influence of indigenous disturbances, surface geometry and roughness, sound, heat transfer, and ablation, it is not possible to develop general prediction schemes for transition location and the nature of turbulent structures in bQundary-layer flows.
There have been a number of recent advances in the mathematical theory of chaos that have been applied to closed systems. Sreenivasan and Strykowski (1984), among others, discuss the extension of these ideas to open systems and conclude that the relationship is still uncertain. It appears from a recent workshop and panel discussion (Liepmann et al. 1986) that the direct application of chaos theory to open systems is still some distance away. However, the prospect of incorporating some of the mathematical ideas of chaos Into open system problems and of encouraging the transfer of data to the mathematicians is good. Since there is still some uncertainty in the direct application of chaos theory to transition no further mention of this will be given here. The purpose of this report is to bring into perspective certain advances to our understanding of laminar-turbulent transition that have occurred within the last five years. In particular, these advances have been made by simultaneous experimental, theoretical, and computational efforts. 1.1 Basic Ideas of Transition With the increased interest in turbulent drag reduction and in large scale structures within the turbulent boundary layer, researchers in turbulence have been required to pay attention to the nature of lam~nar-turbulent transition processes. It is generally accepted that the transition from laminar to turbulent flow occurs because of an incipient instability of the basic flow field. This instability intimately depends on subtle, and sometimes obscure, details of the flow. The process of transition for boundary layers in external flows can be qualitatively described using the following (albeit, oversimplified) scenario. Disturbances in the freestream, such as sound or vorticity, enter the boundary layer as steady and/or unsteady fluctuations of the basic state. This part of the process is called receptivity (Morkovin, 1969) and, although it is still not well understood, it provides the vital initial conditions of amplitude, frequency and phase for the breakdown of laminar flow. Initially these disturbances may be too small to measure and they are observed only after the onset of an instability. The type of instability that occurs depends on Reynolds number, wall curvature, sweep, roughness, and initial conditions. The initial growth of these disturbances is described by linear stability theory. This growth is weak, occurs over a viscous length scale, and can be modulated by pressure gradients, mass flow, temperature gradients, etc. As the amplitude grows three-dimensional and nonlinear interactions occur in the form of secondary instabilities. Disturbance growth is very rapid in this case (now over a convective length scale) and breakdown to turbulence occurs. For many years, linear stability theory, with the Orr-Sommerfeld equation as its keystone, served as the basic tool for predictors and designers. Since the initial growth is linear and its behavior can be easily calculated, transition prediction schemes are usually based on linear theory. However, since the initial conditions (receptivity) are not generally known, only correlations are possible and, most importantly, these correlations must be between two systems with similar environmental conditions. The impossibility of matching or fully understanding these environmental conditions has led to the failure of any absolute transition prediction scheme for even the simple Blasius flat-plate boundary layer. The preceding does not always follow the observed behavior. At times, the initial instability can be so strong that the growth of linear disturbances is by- passed (Morkovin, 1969) in such a way that turbulent spots appear or secondary instabilities occur and the flow quickly becomes turbulent. This phenomenon is not well understood but has been documented in cases of roughness and high freestream turbulence (Reshotko, 1986~. In this case, transition prediction schemes based on linear theory fail completely. 1.2 Review of the Literature The literature review follows the outline of the process described above and begins with Reshotko (1984a, 1986) on receptivity (i.e. the means by which freestream disturbances enter the boundary layer). In these papers, Reshotko summarizes the recent work in this area and points out the difficulties in understanding the problem. Indeed, the receptivity question and the knowledge of the initial conditions are the key issues regarding a transition prediction scheme. Of particular concern to the transition problem are the quantitative details of the roles of freestream sound and turbulence. Aside from some general correlations, this is still an opaque area. However. in section 3.2 below, a demonstration of the role of initial conditions on the observed transition phenomenon is discussed. The details of linear stability theory are given in Mack (1984b). This is actually a monograph on boundary-layer stability theory and should be considered required reading for those interested in all aspects of the subject. It covers 58 pages of text with 170 references. In particular, his report updates the three-dimensional (3-D) material in Mack (1969), covering In large part Mack's own contributions to the area. The foundation paper with regard to nonlinear instabilities is Klebanoff et al. (1962~. This seminal work spawned numerous experimental and theoretical works (not all successful) for the period of 20 years after its publication. It was not until the experimental observations of subharmonic instabilities by Kachanov et al. (1977), Kachanov and Levchenko (1984), and Saric and Thomas (1984), along with the work on secondary instabilities, that additional progress was made in this area. Recent papers of Herbert (1984a,b,c 1985; 1986a,b) cover the problems of secondary instabilities and nonlinearities, i.e. those aspects of the breakdown process that succeed the growth of linear disturbances. It should be emphasized that two- dimensional waves do not completely represent the breakdown process since the transition process is always three-dimensional in bounded shear flows. Herbert describes the recent efforts in extending the stability analysis into regions of wave interactions that produce higher harmonics, three-dimensionality, subharmonics, and large growth rates--all harbingers of transition to turbulence. Recent 3-D Navier-Stokes computations by Fasel (1980,1986), Spalart (1984), 24
Spalart and Yang (1986), Kleiser and Laurien (1985, 1986) Reed and co-workers (Singer et al. 1986, 1987; Yang et al. 1987) have added additional understanding to the phenomena. More is said about this in section 3.2. The paper by Arnal (1984) is an extensive description and review of transition prediction and correlation schemes for two-dimensional flows that covers 34 pages of text and over 100 citations. An analysis of the different mechanisms that cause transition such as Tollmien-Schlichting (T-S) waves, Gortler vortices, and turbulent spots is given. The effects that modulate the transition behavior are presented. These include the influence of freestream turbulence, sound, roughness, pressure gradient, suction, and unsteadiness. A good deal of the data comes from the work of the group at ONERA/CERT part of which has only been available in report form. The different transition criteria that have been developed over the years are also described which gives an overall historical perspective of transition prediction methods. In a companion paper, Poll (1984b) extends the description of transition to 3-D flows. When the basic state is three-dimensional, not only are 3-D disturbances important, but completely different types of instabilities can occur. Poll concentrates on the problems of leading-edge contamination and crossflow vortices, both of which are characteristic of swept-wing flows. The history of these problems as well as the recent work on transition prediction and control schemes for 3-D flows are discussed by Reed and Saric (1989). Reshotko (1984b, 1985, 1986) and Saric (1985b) review the application of stability and transition information to problems of drag reduction and in particular, laminar flow control. They discuss a variety of the laminar flow control and transition control issues which will not be covered here. 2. Review of T-S Waves The disturbance state is restricted to two dimensions with a one-dimensional basic state. The 2- D instability to be considered is a viscous instability in that the boundary-layer velocity profile is stable in the inviscid limit and thus, an increase in viscosity (a decrease in Reynolds number) causes the instability to occur in the form of 2-D traveling waves called T-S waves. All of this is contained within the framework of the Orr-Sommerfeld equation, OSE. The historical development of this work is given in Mack (1984b) and a tutorial is given by Saric (1985a). The OSE is linear and homogeneous and forms an eigenvalue problem which consists of determining the wavenumber, a, as a function of frequency, ce, Reynolds number, R. and the basic state, U(y). The Reynolds number is usually defined as R = UoL/v = SIR,` and is used to represent *distance along the surface. In general, L = Vex /UO is the most straightforward reference length to use because of the simple form of R and because the Blasius variable is the same as y in the OSE. When comparing the solutions of the OSE with experiments, the dimensionless frequency, F. is introduced as F = m/R = 2~fv/Uo: where f is the frequency in Hertz. Usually, an experiment designed to observe T-S waves and to verify the 2-D theory is conducted in a low-turbulence wind tunnel (u'/UO from 0.02% to 0.06%) on a flat plate with zero pressure gradient (determined from the shape factor = 2.59 and not from pressure measurements!) where the virtual-leading- edge effect is taken into account by carefully controlled boundary-layer measurements. Disturbances are introduced by means of a 2-D vibrating ribbon using single-frequency, multiple-frequency, step-function, or random inputs (Pupator and Saric, 1989) taking into account finite-span effects (Mack, 1984a). Hot wires measure the U + u' component of velocity in the boundary layer and d-c coupling separates the mean from the fluctuating part. The frequency, F. for single- frequency waves remains a constant. When the measurements of are repeated along a series of chordwise stations, the maximum amplitude of the waves varies. At constant frequency, the disturbance amplitude initially decays lintil the Reynolds number at which the flow first becomes unstable is reached. This point is called the Branch I neutral stability point and is given by R~. The amplitude grows exponentially until the Branch II neutral stability point is reached which is given by Ru. The locus of R~ and Rn points as a function of frequency gives the neutral stability curve. If the growth rate of the disturbances is defined as c' = o(R,F), Fig. 1 is the locus of o(R,F) =0. For R > 600 the theory and experiment agree very well for Blasius flow. For R < 600 the agreement is not as good because the theory is influenced by nonparallel effects and the experiment is influenced by low growth rates and nearness to the disturbance source. Virtually all problems of practical interest have R > 1000 in which case the parallel theory seems quite adequate (Gaster, 1974; Saric and Nayfeh, 1977~. By assuming that the growth rate, c, = o(R,F), to hold locally (within the quasi-parallel flow approximation), the disturbance equations are integrated along the surface with R = it(x) to give: A/Ao = exp(N) where dN/dR = cs, A and Ao are the disturbance amplitudes at R and R~, respectively, and R~ is the Reynolds number at which the constant-frequency disturbance f~rst becomes unstable (Branch I of the neutral stability curve). The basic design tool is the correlation of N with transition Reynolds number, RT, for a variety of observations. The correlation will produce a number for N (say 9) which is now used to predict RT for cases in which experimental data are not available. This is the celebrated eN method of Smith and von Ingen (e.g. Arnal, 1984; Mack, 1984b). The basic LFC technique changes the physical parameters and keeps N within reasonable limits in order to prevent transition. As long as laminar flow is maintained and the disturbances remain linear, this method contains all of the necessary physics to accurately predict disturbance behavior. As a transition prediction device, the eN method is certainly the most popular technique used today. It works within 25
some error limits only if comparisons are made with experiments with identical disturbance environments. Since no account can be made of the initial disturbance amplitude this method will always be suspect to large errors and should be used with extreme care. When bypasses occur, this method does not work at all. Mack (1984b) and Arnal (1984) give examples of growth-rate and eN calculations showing the effects of pressure gradients, Mach number, wall temperature, and three dimensionality for a wide variety of flows. These reports contain the most up-to-date stability . ~ . ntormatlon. 3. Secondary Instabilities and Transition There are different possible scenarios for the transition process, but it is generally accepted that transition is the result of the uncontrolled growth of unstable three-dimensional waves. Secondary instabilities with T-S waves are reviewed in some detail by Herbert (1984b, 1985, 1986), Saric and Thomas (1984) and Saric et al.(l984~. Therefore, only a brief outline is given in section 3.1 in order to give the reader some perspective of the different types of breakdown. Section 3.2 discusses the very recent results. 3.1 Secondary Instabilities The occurrence of three-dimensional phenomena in an otherwise two-dimensional flow is a necessary prerequisite for transition (Tan), 1981~. Such phenomena were observed in detail by Klebanoff et al. (1962) and were attributed to a spanwise differential amplification of T-S waves through corrugations of the boundary layer. The process leads rapidly to spanwise alternating "peaks" and "valleys", i.e., regions of enhanced and reduced wave amplitude, and an associated system of streamwise vortices. The peak- valley structure evolves at a rate much faster than the (viscous) amplification rates of T-S waves. The schematic of a smoke-strealdine photograph (Saric et al. 1981) in Fig. 1 shows the sequence of events after the onset of "peak-valley splitting". This represents the path to transition under conditions similar to Klebanoff et al. (1962) and is called a K-type breakdown. The lambda-shaped (Hama and Nutant, 1963) spanwise corrugations of streaklines, which correspond to the peak-valley structure of amplitude variation, are a result of weak 3-D displacements of fluid particles across the critical layer and precede the appearance of Klebanoff's "hair-pin" vortices. This has been supported by hot- wire measurements and Lagrangian-type streakline prediction codes (Saric et al.,l981; Herbert and Bertolotti, 1985~. Note that the lambda vortices are ordered in that peaks follow peaks and valleys follow valleys. Since the pioneering work of Nishioka et al.(l975, 1980), it is accepted that the basic transition phenomena observed in plane channel flow are the same as those observed in boundary layers. Therefore, little distinction will be given here as to whether work was done in a channel or a boundary layer. From the theoretical and computational viewpoint, the plane channel is particularly convenient since the Reynolds number is constant, the mean flow is strictly parallel, certain symmetry conditions apply, and one is able to 26 do temporal theory. Thus progress has been first made with the channel flow problem. Different types of three-dimensional transition phenomena recently observed (e.g. Kachanov et al. 1977; Kachanov and Levchenko, 1984; Saric and Thomas, 1984; Saric et al. 1984, Kozlov and Ramanosov, 1984) are characterized by staggered patterns of peaks and valleys (see Fig. 2) and by their occurrence at very low amplitudes of the fundamental T-S wave. This pattern also evolves rapidly into transition. These experiments showed that the subharmonic of the fundamental wave (a necessary feature of the staggered pattern) was excited in the boundary layer and produced either the resonant wave interaction predicted by Craik (1971) (called the C- type) or the secondary instability of Herbert (1983) (called the H-type). Spectral broadening to turbulence with self-excited subharmonics has been observed in acoustics, convection, and free shear layers and was not identified in boundary layers until the results of Kachanov et al. (1977~. This paper reinitiated the interest in subharmonics and prompted the simultaneous verification of C-type resonance (Thomas and Saric, 1981; Kachanov and Levchenko, 1984~. Subharmonics have also been confirmed for channel flows (Kozlov and Ramazanov, 1984) and by direct integration of the Navier-Stokes equations (Spalart 19841. There is visual evidence of subharmonic breakdown before Kachanov et al. (1977) in the work of Hama (1959) and Knapp and Roache (1968) which was not recognized as such at the time of their publication. The recent work on subharmonics is found in Herbert (1985, 1986a,b), Saric, Kozlov and Levchenko (1984), and Thomas (1986~. The important issues that have come out of the subharmonic research is that the secondary instability depends not only on disturbance amplitude, but on phase and fetch as well. Fetch means here the distance over which the T-S wave grows in the presence of the 3-D background disturbances. If T-S waves are permitted to grow for long distances at low amplitudes subharmonic secondary instabilities are initiated at disturbance amplitudes of less than 0.3%Uo. Whereas, ~f larger amplitudes are introduced, the breakdown occurs as K-type at amplitudes of 1% UO. Thus, there no longer ex~sts a "magic" amplitude criterion for breakdown. A consequence of this requirement of a long enough fetch for the subharmonic to be entrained from the background disturbances is that the subharmonic interaction will occur at or to the right of the Branch II neutral stability point. Since this is in the stable region of the fundamental wave, it was not likely to be observed because the experimenters quite naturally concentrated their attention of measurements between Branch I and Branch II. 3.2 Recent Results The surprise that results from the analytical model of Herbert (1986a,b) and the Navier-Stokes computations of Singer, Reed, and Ferziger (1986), is that under conditions of the experimentally observed K- Type breakdown, the subharmonic H-Type is still the dominant breakdown mechanism instead of the
fundamental mode. This is in contrast to Klebanoff's experiment, confirmed by Nishioka et al. (1975, 1980),Kachanov et al. (1977), Saric and Thomas (1984), Saric et al. (1984), and Kozlov and Ramazanov (1984) where only the breakdown of the fundamental into higher harmonics was observed. Only Kozlov and Ramazanov (1984) observed the H-type in their channel experiments and only when they artificially introduced the subharmonic. This apparent contradiction was resolved by Singer, Reed, and Ferziger (1987~. Here the full three- dimensional, time-dependent incompressible Navier- Stokes equations are solved with no-slip and impermeability conditions at the walls. Periodicity was assumed in both the streamwise and spanwise directions. The implementation of the method and its validation are described by Singer, Reed and Ferziger (19861. Initial conditions include a two-dimensional T- S wave, random noise, and streamwise vortices. No shape assumptions are necessary, the spectrum is larger, and random disturbances whether freestream or already in the boundary layer can be introduced and monitored for growth and interactions (Singer, Reed, and Ferziger, 1986~. Other advantages realized by computations are 1) the inclusion of boundary-layer growth, neglected in linear theory but important to the growth of secondary instabilities, 2) the generation of ensemble averages, 3) the visualization of flow phenomena for comparison with experiments (advanced graphics capability), and 4) the calculation of vorticity and energy spectra, often unavailable from experiments. The streamwise vortices can alter the relative importance of the subharmonic and fundamental modes. Streamwise vortices of approximately the strength of those that might be found in transition experiments can explain the difficulty in experimentally identifying the subharmonic route to turbulence (Herbert 1983~. The corresponding computational visualizations of Singer et al. (1987) are shown in Figs. 3 and 4; flow is from lower right to upper left. Figure 4 shows the vortex structures, commonly seen in the transition process, under the conditions of a forced 2-D T-S wave and random noise as initial conditions. The subharmonic mode is present as predicted by theory but not seen experimentally. Other views of the vertical structure are given by Herbert (1986a). However, when streamwise vorticity (as is present in the flow from the turbulence screens upstream of the nozzle) is also included, the subharmonic mode is overshadowed by the fundamental mode (as in the experiments!. The resulting pattern, ordered peak-valley structure, is seen in Fig. 4. Here is a case in which the computations have explained discrepancies between theory and experiments. In the presence of streamwise vorticity, the fundamental mode is preferred over the subharmonic; this agrees with experimental observations, but not with theory (which does not account for this presence). Without streamwise vorticity, the subharmonic modes dominate, as predicted by theory and confirmed by computational simulations. In the presence of streamwise vorticity characteristic of wind-tunnel experiments, the K-type instability dominates and the numerical simulations predict the experimental results. Direct numerical simulations are playing an increasingly important role in the investigation of transition; the literature is growing, especially recently. This trend is likely to continue as considerable progress is expected towards the development of new, extremely powerful supercomputers. In such simulations, the full Navier-Stokes equations are solved directly by employing numerical methods, such as finite-difference or spectral methods. The direct simulation approach is widely applicable since it avoids many of the restrictions that usually have to be imposed in theoretical models. The Navier-Stokes solutions are taken hand-in- hand with the wind tunnel experiments in a complementary manner. The example of Singer, Reed, and Ferziger (1987) illustrates that these two techniques cannot be separated. The next step in the simulations will be to predict the growing body of detailed data being developed by Nishioka et al. (1980, 1981, 1984, 1985) on the latter stages of the breakdown process. 4 Transition Prediction and Control . When the recent work on subharmonics is added to the discussion at the end of section 3 on the limitations of the eN method, one indeed has an uncertainty principle for transition (Morkovin, 1978~. Transition prediction methods will remain conditional until the receptivity problem is adequately solved and the bypass mechanisms are well understood. In the mean time, extreme care must be exercised when using correlation methods to predict transition. Additional problems of transition prediction and laminar flow control are discussed by Reshotko (1985, 1986~. The main principle of laminar flow control is to keep the disturbance levels low enough so that secondary instabilities and transition do not occur. Under these conditions, linear theory is quite adequate and eN methods can be used to calculate the effectiveness of a particular LFC device. The idea of transition control through active feedback systems is an area Hat has received considerable recent attention (Liepmann and Nosenchuck, 1982; Thomas, 1983; Kleiser and Laurien, 1984, 1985; Metcalfe et al., 1985~. The technique consists of first sensing the amplitude and phase of an unstable disturbance and then introducing an appropriate out-of-phase disturbance that cancels the original disturbance. In spite of some early success, this method is no panacea for the transition problem. Besides the technical problems of the implementation of such a system on an aircraft, the issue of three- dimensional wave cancellation must be addressed. As Thomas (1983) showed, when the 2-D wave is canceled, all of the features of the 3-D disturbances remain to cause transition at yet another location. Some clear advantages over passive systems have yet to be demonstrated for this technique. Acknowledgements This work is supported by the Air Force Office of Scientific Research Contract AFOSR-85-NA-077. 27
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- / Off ~ f Figure 1. Staggered peak-valley structure 01-type mode). / f f f f > f Figure 2. Peak-valley splitting structure (K-type mode). 31
amp Figs ~ Vortex lines with ~ = ~0~, f = 177. The Dow goes ham lower tight to upper lea. Figure 4. Vo~x lees with ~ = ~0~, ~ = 132. Tbo Cow goes Mom lower tight to upper Ad. ElUpOcal s~c~wisc voices with maximum ~ pc~urbadon of 1 8~ arc included. 32
DISCUSSION by F. Stern Please comment on the influence of the type of breakdown on the state of the resulting turbulent boundary layer, which is quite important both in experiments and calculations. Also, are there any similarities between the processes you have classified for transition and those associated with relaminarization? Author's Reply The first part of this question hits to the heart of the motivation for doing work on the latter stages of transition. The type of transition is important because of its influence on the large-scale structure in "low" Reynolds number turbulent boundary layers. The control of turbulent boundary layers then rests on the type of large scale structure that may be present. For "high" Reynolds number turbulent boundary layers, the situation is not so clear i.e. it is hard to imagine that the details of the transition process influence the structure of fully developed turbulence. It is unlikely that any of the structure of the transition process is recovered during relaminarization of a turbulent flow. I believe relaminarization to be highly dissipative due to large changes in the basic state which cause a loss in the turbulence production mechanisms. Unfortunately, there is a dearth of detailed experiments in this area. 33