Laminar-turbulent transition prediction in three-dimensional flows

Progress in Aerospace Sciences 36 (2000) 173}191

Laminar-turbulent transition prediction in three-dimensional #ows D. Arnal*, G. Casalis ONERA Centre de Toulouse, DMAE/TRIN, BP 4025-2 avenue Edouard Belin, 31055 Toulouse Cedex-France

Abstract Accurate laminar-turbulent transition prediction is needed for many practical problems. This paper presents an overview of the methods which can be used today to estimate the location of the transition onset. The applications are restricted to subsonic #ows (incompressible and transonic) developing on swept wings of in"nite span. It is assumed that the environmental disturbances are small, so that the transition process is governed by the spatial growth of unstable eigenmodes. Therefore the main theoretical tools are stability theories, linear and nonlinear. The most popular prediction method is the e, method which is based on linear theory. This method was initially developed for two-dimensional #ows and then extended to more complex problems. The paper describes various approaches currently available to compute the N factor; examples of application are given for wind tunnel and free #ight experiments. More sophisticated prediction methods make use of weakly nonlinear theories, the governing equations being solved by a PSE (Parabolized Stability Equations) approach. In this case, the problem of choosing a critical value of the N factor at transition is replaced by the problems of de"ning the most interesting nonlinear scenario and of imposing adequate initial conditions. The examples of application presented in the paper illustrate the need for a better understanding of the receptivity mechanisms.  2000 Elsevier Science Ltd. All rights reserved.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2. Qualitative description of transition mechanisms 2.1. Two-dimensional #ows . . . . . . . . . . . . . 2.2. Three-dimensional #ows . . . . . . . . . . . . 3. Theoretical background . . . . . . . . . . . . . . . 3.1. Linear stability theory: local approach . . . 3.1.1. Assumptions. Eigenvalue problem . . 3.1.2. Some important results . . . . . . . . . 3.1.3. TS and CF instabilities . . . . . . . . . 3.2. Linear stability: non-local approach . . . . . 3.3. Receptivity . . . . . . . . . . . . . . . . . . . . 3.4. Non-linear interactions . . . . . . . . . . . . .

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* Corresponding author. Fax: 33-5-62-25-25-83. E-mail addresses: [email protected] (D. Arnal); [email protected] (G. Casalis). 0376-0421/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 6 - 0 4 2 1 ( 0 0 ) 0 0 0 0 2 - 6

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4. Prediction methods based on linear theories . . . . . . . . . 4.1. Local approach . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Two-dimensional #ows . . . . . . . . . . . . . . . 4.1.2. Three-dimensional #ows . . . . . . . . . . . . . . 4.1.3. Shortcomings of the local e, method . . . . . . . 4.2. Non-local approach . . . . . . . . . . . . . . . . . . . . . 4.3. Comparison between local and non-local results . . . . 4.4. Simpli"ed stability methods . . . . . . . . . . . . . . . . 5. Non-linear approaches . . . . . . . . . . . . . . . . . . . . . . 5.1. Weakly non-linear instabilities with the PSE approach 5.1.1. Two-dimensional and streamwise instabilities . . 5.1.2. Cross#ow instability . . . . . . . . . . . . . . . . 5.2. Practical use of non-linear PSE for swept wing #ows . 5.2.1. Scenarios . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Initial amplitudes . . . . . . . . . . . . . . . . . . 5.3. High-frequency secondary instability . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction Since the classical experiments performed by Osborne Reynolds (1883), the instability of laminar #ow and the transition to turbulence have maintained a constant interest in #uid mechanics problems. This interest results from the fact that transition controls important aerodynamic quantities such as drag or heat transfer. For example, the heating rates generated by a turbulent boundary layer may be several times higher than those for a laminar boundary layer, so that the prediction of transition is of great importance for hypersonic reentry spacecraft, because the thickness of the thermal protection is strongly dependent upon the altitude where transition occurs. For high-subsonic speed, commercial transport aircraft, the achievement of laminar #ow can signi"cantly reduce the drag on the wings and hence the speci"c consumption of the aircraft. The potential bene"ts are important, because transition separates the laminar #ow region with low drag from the turbulent region where skin friction dramatically increases. A good knowledge of the transition mechanisms is also required to reduce hydrodynamically generated noise beneath transitional #ows. These examples explain why for many years a large amount of theoretical, numerical and experimental work has been devoted to the laminar-turbulent transition problems. It is now established that transition is the result of a sequence of complex phenomena which depend on many parameters, such as Reynolds number, pressure gradient, disturbance environment, etc. There are several routes to turbulence, which can be summarized as follows. Let us consider a laminar #ow developing along a given body. The "rst stage of the transition process is the boundary layer receptivity [1]. Receptivity describes the means by which the environmental disturbances (noise, free stream turbulence, vibrations, etc.)

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enter the laminar boundary layer, as well as their signature in the disturbed #ow. This signature constitutes the initial conditions for the development of more or less complex mechanisms which ultimately lead to turbulence. After the forced disturbances have been internalized by the boundary layer, two kinds of transition processes can be distinguished: E If the amplitude of the forced disturbances is small, one can observe the appearance of more or less regular oscillations which start to develop downstream of a certain critical point. These waves constitute the eigenmodes of the laminar boundary layer and the "rst stage of their evolution can be described by a linear theory. They are ampli"ed up to the point where transition occurs (breakdown). This process is called natural transition. E If the amplitude of the forced disturbances is not weak (high free stream turbulence level, large isolated roughness elements), nonlinear phenomena are immediately observed and transition occurs rapidly. This mechanism is called a bypass in the sense that the linear stages of the transition process do not occur (they are `bypasseda). It is clear that a rigorous modelling of the transition process is a very di$cult task. A pessimistic conclusion could be that a `gooda prediction of transition is impossible. In spite of this negative situation, transition predictions must be performed for many industrial problems. The goal of this paper is to provide an overview of the theoretical and numerical techniques which are currently used to predict the onset of transition. Emphasis will be given on three-dimensional #ow problems, which are of major interest for many practical problems (laminar #ow control for instance). Due to space limitation, the discussion will be restricted to `natural transitionsa

D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

175

occurring in subsonic yows (incompressible and transonic). Section 2 gives a qualitative description of the main mechanisms leading to turbulence. In the case of `natural transitiona considered here, the linear stability theory can be used to determine the characteristics of the unstable disturbances mentioned before. This theory, as well as some other theoretical elements dealing with receptivity and nonlinear phenomena, are presented in Section 3. Sections 4 and 5 describe the transition prediction methods based on linear and on nonlinear approaches, respectively.

Fig. 1. Laminar boundary layer development on a swept wing. x is the location of the inviscid streamline in#ection point. b is +  the angle between wall and potential streamlines.

2. Qualitative description of transition mechanisms

2.2. Three-dimensional yows

2.1. Two-dimensional yows

Let us consider the case of a laminar boundary layer developing on a swept wing. As it is illustrated in Fig. 1, the mean velocity pro"le is now decomposed into a streamwise velocity proxle u (in the direction of the external streamline) and a crossyow velocity proxle w (in the direction normal to this streamline). From the leading edge to the chordwise location x where the free stream velocity is maximum, the cross+ #ow is directed towards the concave part of the external streamline. Its magnitude is zero at the attachment line, then it increases more or less rapidly due to the #ow acceleration. In the region of negative pressure gradient, the maximum value of the cross#ow velocity component remains rather weak, about 5}10% of the free stream velocity; it will be shown, however, that this is su$cient to create a strong cross#ow instability. As x is approached, the intensity of the pressure + gradient decreases, leading to a decrease in the cross#ow amplitude. At x"x , the pressure gradient becomes + positive, the curvature of the external streamline changes and the velocity close to the wall reverses (S-shaped pro"les). If the positive pressure gradient is strong enough, the cross#ow velocity pro"le can be completely reversed. In the same region, an in#ection point appears on the streamwise pro"le u. The basic phenomena are qualitatively the same for two- and three-dimensional #ows. The major di!erence is that in the latter case the unstable waves now develop in a very wide range of propagation directions. The linear stability theory shows that the most unstable propagation direction strongly depends on whether or not the #ow is accelerated or decelerated. A peculiarity of three-dimensional #ow instability is that zero frequency disturbances become highly unstable as soon as the cross#ow velocity component is large enough. In the experiments, these stationary disturbances can be visualized as more or less regularly spaced streaks practically aligned in the streamwise direction. A large amount of experimental work has been devoted during the last years to the analysis of nonlinear

Let us consider "rst the simple case of two-dimensional #ows. As stated before, the instability leading to transition starts with the development of wave-like disturbances, the existence of which was "rst demonstrated by the now classic experiments of Schubauer and Skramstad [2]. The amplitude of these so-called Tollmien-Schlichting (TS) waves exhibits at "rst an exponential growth, which can be computed by the linear stability theory. Further downstream, the disturbances reach a "nite amplitude, so that their development begins to deviate from that predicted by the linear theory. Low speed experiments have shown that the initially twodimensional TS waves are distorted into a series of `peaksa and `valleysa. Further downstream, three-dimensional and nonlinear e!ects become more and more important. The nonlinear development of the disturbances terminates with the `breakdowna phenomenon: experiments and Direct Numerical Simulations (DNS) indicate that the peak-valley structures are stretched and form horseshoe vortices which break down into smaller vortices, which again break down into smaller vortices. The #uctuations "nally take a random character and form a turbulent `spota. The streamwise location where the "rst spots appear can be de"ned as the onset of transition. At this point, the shape factor of the mean velocity pro"le begins to decrease, while the skin friction (and also the wall heat transfer for high speed #ows) starts to exceed its laminar value. The last stage is the development of the turbulent spots up to the fully turbulent regime. In fact, the distance between the end of the linear region and the breakdown to turbulence is rather short: for #at plate conditions (i.e. without pressure gradient), the streamwise extent of the linear ampli"cation covers about 75}85% of the distance between the leading edge and the beginning of transition. This explains why most of the practical transition prediction methods are based on linear stability only.

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D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

mechanisms occurring in three-dimensional #ows, see for instance [3}7]. However, it can be noticed that the extent of the region where nonlinearities play a dominant role is much larger than for two-dimensional #ows, at least when transition is triggered by a pure cross#ow instability.

In this paper, the discussion will be restricted to the so-called spatial theory, see [39], which is more relevant than the temporal theory for boundary layer #ows. u is real, a and b are complex: a"a #ia and b"b #ib . P G P G r is expressed by r"r(y) exp(!a x!b z) exp[i(a x#b z!ut)]. G G P P

3. Theoretical background

It is now possible to de"ne a wavenumber vector k"(a , b ) and an ampli"cation vector A"(!a ,!b ), P P G G with angles t and tK with respect to the x direction:

3.1. Linear stability theory: local approach A comprehensive description of this theory can be found in [8]. 3.1.1. Assumptions. Eigenvalue problem The principle of linear stability theory is to introduce small sinusoidal disturbances into the Navier}Stokes equations in order to compute the range of unstable frequencies. It is assumed that any #uctuating quantity r (velocity, pressure, density or temperature) is expressed by: r"r(y) exp[i(ax#bz!ut)]

(2)

(1)

x, y, z is an orthogonal coordinate system, which can be either cartesian or curvilinear, y being normal to the surface. The complex amplitude function r depends on y only. In the general case, a, b and u are complex numbers. The #uctuating quantities are very small, so that the quadratic terms of the disturbances can be neglected in the Navier}Stokes equations. It is also assumed that the mean #ow quantities do not vary signi"cantly over a wavelength of the disturbances; therefore u and w (mean #ow components in the x and z directions) as well as the mean temperature ¹ are functions of y only, and the normal velocity v is equal to zero. The implication of this parallel yow approximation is that the stability of the #ow at a particular station (x, z) is determined by the local conditions at that station independently of all others. This leads to a system of homogeneous, ordinary differential equations for the amplitude functions r(y). For the simplest case of a two-dimensional, low speed #ow with b"0, the stability equations can be combined to obtain a single equation for the vertical component v of the velocity #uctuation. This is the well known Orr} Sommerfeld equation. Due to the homogeneous boundary conditions (the disturbances must vanish at the wall and in the free stream, except the pressure #uctuations which have a non-zero amplitude at the wall), the problem is an eigenvalue one: when the mean #ow is speci"ed, nontrivial solutions exist only for certain combinations of the parameters a, b, u and R (Reynolds number). This constitutes the dispersion relation.

t"tan\(b /a ), P P

(3)

tK "tan\(b /a ). G G

(4)

If b is set equal to zero, the waves can be ampli"ed G (a (0), neutral (a "0) or damped (a '0) in the x G G G direction. From these de"nitions, it is now clear that any eigenvalue problem involves six real parameters (a , a , b , b , P G P G u, R). This set of parameters is often replaced by (a , a , t, P G tK , u, R). Numerically, the input data are the mean velocity and mean temperature pro"les, the free stream Mach number, the stagnation temperature, the #uid properties and all but two of the previous real parameters. The computation gives the values of the two remaining parameters (eigenvalues) as well as the disturbance amplitude pro"les (eigenfunctions). Some examples of the numerical procedures are described in [8]. 3.1.2. Some important results Let us consider "rst incompressible #ows (M "0, C where M is the free stream Mach number). In the frameC work of the so-called inviscid theory, it is assumed that the viscosity acts only on the establishment of the mean velocity pro"le. Neglecting the terms of the order 1/R in the stability equations leads to the Rayleigh equation. As this equation is much simpler than the Orr}Sommerfeld equation, a number of mathematical results have been obtained (see overview in Drazin and Reid [9]). It has been shown that a necessary condition for an instability of any inviscid shear #ow is that the #ow must have an in#ection point and that the extremum of vorticity du/dy associated with this point must be a minimum (Rayleigh and Fjortoft theorems). It has also been shown that this condition is su$cient for bounded shear #ows. Because this inviscid instability is strong, the appearance of in#ectional pro"les usually implies a rapid breakdown to turbulence. The extension of the inviscid theory to compressible #ows has been carried out by Lees and Lin [10]. In this theory, the in#ection point is replaced by the generalized in#ection point, which is the point where




du d o "0. dy dy

D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

Applying the compressible theory to the low Mach number case illustrated here, the generalized in#ection point is very close to the classical in#ection point, so that the previous results still apply in a "rst approximation. It is well known, however, that a velocity pro"le without an in#ection point (Blasius pro"le for instance) is likely to be unstable. In this case, viscosity establishes the no-slip condition at the wall, which in turn creates a Reynolds stress that may destabilize the #ow. This instability is called viscous because the boundary layer is stable in the inviscid limit, but a decrease in the Reynolds number (i.e. an increase in viscosity) causes the instability. For two-dimensional #ows, a distinction is usually made between two-dimensional waves (with crests normal to the mean #ow direction) and oblique waves (often referred to as three-dimensional waves). The Squire theorem tells us that a two-dimensional wave starts to be ampli"ed at a lower Reynolds number than any threedimensional wave. This theorem holds only for incompressible #ows with temporally growing disturbances. It does not imply that a two-dimensional wave is always more ampli"ed than a three-dimensional wave. An interesting result can be deduced when the assumptions and the mathematical transformation used by Squire are applied to three-dimensional mean #ows. It can be demonstrated that the temporal stability problem in any direction t reduces to a two-dimensional problem for the velocity pro"le projected in this direction. In other words, a three-dimensional problem can be considered as a series of two-dimensional problems for all possible values of t (Stuart, in [11]). 3.1.3. TS and CF instabilities For a given mean #ow "eld and for "xed values of the t and tK angles, it is possible to compute a stability diagram as shown in the upper part of Fig. 2. This diagram gives the range of unstable frequencies f"u/2p as a function of the streamwise distance. A neutral curve separates the region where the waves are ampli"ed (inside the neutral loop) from the region where the waves are damped (outside the neutral loop). For each frequency, there is a critical abscissa x below which the  disturbances are damped. In the case of two-dimensional zero pressure gradient #ows, the shape of the stability diagrams does not change very much from incompressible to transonic #ow. An interesting feature is that the growth rates are smaller in the transonic range than for incompressible #ow: compressibility has a stabilizing e!ect. Another important aspect of instability for transonic Mach numbers is the e!ect of the wavenumber direction t on the ampli"cation rates. Up to Mach numbers of the order of 0.6}0.8, the maximum value of !a corresponds to t"03 (for G a "xed frequency at a given Reynolds number). At higher Mach numbers, the largest ampli"cation rates are

177

Fig. 2. Typical stability diagram in physical coordinates. De"nition of the total growth rate and of the envelope curve.

obtained for non-zero values of t (oblique waves). Typically, the most unstable direction is around 40 or 503 for M close to 1. These results remain valid for non-zero C pressure gradient #ows. In decelerated #ows, the appearance of an in#ection point generates an inviscid instability which can play a role even at low Reynolds numbers. Accelerated #ows make the #ow unstable at large Reynolds number through the action of viscous instability. The mechanisms are more complex for three-dimensional mean #ows but they are now relatively well understood, see [12}15]. As a "rst approximation, it can be assumed that transition is triggered either by streamwise instability (or TS instability) or by crossyow instability (CF instability): E Streamwise instability: as the streamwise mean velocity pro"les u look like classical two-dimensional pro"les, they are essentially unstable in decelerated #ows (positive pressure gradients), i.e. downstream of point x in Fig. 1. The wavelengths of the streamwise dis+ turbances are about 10 times the boundary layer thickness. E Cross#ow instability: as an in#ection point is always present in the cross#ow mean velocity pro"le w, a powerful in#ectional instability is expected to occur in regions where the cross#ow velocity develops rapidly. This phenomenon is observed, for instance, in the vicinity of the leading edge of a swept wing, i.e. upstream of point x in Fig. 1. The most unstable distur+ bance propagation direction t (relative to the + external streamline) is never exactly equal to 903. It lies in a narrow range close to the cross#ow direction z, say between 853 and 893. The instability is dominated

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by the properties of the in#ection point (altitude, local value of the shear stress dw/dy...) which are not very much a!ected by compressibility e!ects. The wavelengths of the cross#ow disturbances are 3 to 4 times the boundary layer thickness. Linear stability theory shows that CF instability can amplify zero frequency disturbances. As explained in Section 2, this leads to the formation of stationary vortices, the axes of which are close to the streamwise direction. Immediately downstream of the point of maximum free stream velocity, CF disturbances are decaying. This is due to the fact that the absolute value of the cross#ow velocity of the S-shaped pro"les is low, so that the shear stress at the in#ection point of these pro"les is small. If the boundary layer is still laminar, TS disturbances start to be ampli"ed in this region. 3.2. Linear stability: non-local approach A new formulation for the stability analysis was independently proposed by Herbert [16] and by Dallmann at DLR GoK ttingen [17,18]. By comparison with the local theory, the major improvement is that non-parallel effects are taken into account, as well as the wall curvature e!ects, which are of the same order of magnitude. It is also possible to introduce the non-linear terms, see Section 3.4. In this approach, the expression of the disturbances is r"r(x, y) exp[i(h(x)#bz!ut)] with dh/dx"a(x). (5) The choice of the coordinate system will be discussed in Section 4.2. In the previous equation, a is complex, b and u are real and constant. In contrast to the local approach expressed by relation (1), the amplitude functions depend on y and x, and a depends on x. Substituting the previous expression into the stability equations, neglecting *r/*x and linearizing in r yield a partial di!erential equation of the form: *r da ¸r#M # Nr"0 *x dx

(6)

where ¸, M and N are operators in y with coe$cients that depend on x and y through the appearance of the basic #ow pro"les. When a is computed from a so-called normalization condition, the previous equation can be solved using a marching procedure in x with prescribed initial conditions: this constitutes the PSE (Parabolized Stability Equations) approach. As the results of this approach at a given x station depend on the upstream history of the disturbances, this approach is called non-local. In the frame of the non-local theory, the physical growth rate p of an arbitrary disturbance quantity Q is

de"ned as:

 

1 *Q p"!a #Real . G Q *x

(7)

Usually, Q is taken to be u, v, w, ¹, (ou), either at some "xed y position or at the location where the considered quantity reaches its maximum value. The disturbance kinetic energy can also be used as a measure of the disturbance growth. In the general case, di!erent disturbance quantities exhibit di!erent growth rates, whereas they exhibit the same growth rate in the framework of the local theory. Due to the formal complexity of the PSE equations, it is impossible to deduce simple theorems such as those which have been obtained from the Rayleigh equation. However, as the non-local e!ects represent a "rst-order correction to the local equations, they do not substantially modify the main conclusions of the local, inviscid theory; in particular, the appearance of an in#ection point still remains a major source of instability. 3.3. Receptivity How are the unstable waves excited by the available disturbance environment? This question is a part of the problem which is usually addressed under the term `receptivitya, introduced by Morkovin [1]. Let us recall that receptivity is the process by which disturbances impose their signature on the boundary layer. A part of this signature is the development of the unstable waves described before. For two-dimensional, low-speed #ows, theoretical studies conducted by Goldstein [19] and Kerschen [20] have demonstrated that the receptivity process occurs in regions of the boundary layer where the mean #ow exhibits rapid changes in the streamwise direction. This happens near the body leading edge and/or in any region farther downstream where some local feature forces the boundary layer to adjust on a short streamwise lengthscale (sudden change in the wall slope or in the wall curvature, suction strip for instance). In three-dimensional #ows, the receptivity problem is met in two ways, because stationary and travelling disturbances result from di!erent mechanisms. As indicated above, travelling waves are initiated by non-zero frequency excitations such as free stream turbulence or sound. Careful experiments have been carried out by Deyhle and Bippes [21] on a swept model by varying systematically the disturbance environment in the case of CF dominated transitions. It was observed that low free stream turbulence levels can delay transition by decreasing the growth of the stationary vortices. With higher free stream turbulence levels, travelling waves become dominant and accelerate transition. By contrast with twodimensional #ows, the receptivity to sound was observed

D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

to be weak. As far as stationary waves are concerned, experiments performed by Radetsky et al. [22] and by Deyhle and Bippes [21] demonstrated that three-dimensional, micron-sized roughness elements around a swept leading edge provide the initial amplitude of these vortices. This constitutes another receptivity mechanism which is typical of cross#ow instability. 3.4. Non-linear interactions Today, several approaches are available to analyse the behaviour of the disturbances when they begin to deviate from the linear ampli"cation regime. These numerical tools include, for instance, the method of multiple scales or the secondary instability theories [23]. DNS are of course particularly helpful for improving the understanding of the non-linear mechanisms. For practical applications, nonlinear PSE are particularly attractive; although they only describe weakly non-linear phenomena, they are less time consuming than DNS and they can be used for rather complex geometries. The basic ideas of the non-linear PSE have been explained in many papers, see for instance [16]. The mean #ow is assumed to be independent on the spanwise coordinate z and on the time t. The disturbances are written as double Fourier expansions containing two- and three-dimensional discrete normal modes denoted as (n, m) modes:   r" r (x, y) LK L\ K\ V ;exp i a (m) dm#mbz!nut (8) LK V a is complex, b and u are real and constant. It can be LK seen that n represents the time dependence, and m the spanwise dependence. As has been done in the linear PSE approach, the x-dependence of the (n, m) modes is shared between the amplitude functions and the exponential terms. A normalization condition is still applied in order to ensure the weak chordwise variation of the amplitude functions. Some terms in Eq. (8) have an interesting physical meaning. For example, the (0, m) modes do not depend on time. At a given x position, they distort the basic #ow periodically in the spanwise direction z. The (0, 0) mode is referred to as the `mean #ow distorsiona, which is uniform in the z direction. On the other hand, the (n, 0) modes are time dependent, but independent of z; for two-dimensional #ows, they correspond to two-dimensional waves. The negative values of the integers n and m in the Fourier expansion correspond to complex conjugate values of the positive ones. They are introduced because the considered #uctuations must be real; the relation r "r is thus imposed. High-order terms (large \L\K LK

 



179

values of n and m) correspond to high harmonics of the fundamental mode; they have very low amplitudes in the linear regime, but they grow rapidly in the nonlinear phase, leading to the appearance of high-frequency disturbances with small wavelengths. Introducing a truncated form of (8) (with for instance N "M "5) into the Navier}Stokes equations



 leads to a system of coupled partial di!erential equations which is solved by a marching procedure. Each equation can be written with a left-hand side which exactly coincides with the corresponding linear equation; the right-hand side describes the nonlinear interactions acting on the considered mode. The interactions are usually limited to the quadratic products of two #uctuating terms, so that the terms acting on the (n, m) mode are the (r, s) and (p, q) modes such that r#p"n and s#q"m. The complete set of non-linear PSE equations consists of the previous system of coupled equations, a normalization condition for each mode, boundary conditions and initial conditions. These initial conditions are the amplitude functions r , the streamwise wave numLK bers a and the dimensional amplitude of the modes at LK the initial abscissa. The choice of appropriate initial conditions is often a critical issue for non-linear PSE computations.

4. Prediction methods based on linear theories The so-called e, method is widely used by people who are assigned the job of making transition prediction. It was "rst developed by Smith and Gamberoni [24] and by van Ingen [25] for low-speed #ows and then extended to compressible and/or three-dimensional #ows. This method is based on linear theory only, so that many fundamental aspects of the transition process are not accounted for. However, one has to keep in mind that there is no other practical method presently available for industrial applications. The e, method can be used either with the local (classical) stability approach or with the non-local (PSE) stability approach. Both aspects are examined successively. 4.1. Local approach 4.1.1. Two-dimensional yows For incompressible #ows, only two-dimensional waves (b"0) are considered. The disturbances are ampli"ed or damped according to the sign of the spatial growth rate !a . G The principle of the e, method can be understood from Fig. 2. Let us consider a wave which propagates downstream with a "xed frequency f . This wave passes at "rst  through the stable region. It is damped up to x , then  ampli"ed up to x , and then it is damped again down stream of x . At a given station x, the total ampli"cation 

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rate of a spatially growing wave can be de"ned by



ln(A/A )" 

V

!a dx. (9) G V In the previous expression, A is the wave amplitude, and the index 0 refers to the streamwise position where the wave becomes ampli"ed. The envelope of the total ampli"cation curves is N"Max [ln(A/A )]. (10)  D In incompressible #ow, with a low-disturbance environment, it is assumed that transition occurs as soon as the N factor reaches a critical value in the range 7}10, i.e. when the locally most unstable frequency is ampli"ed by a factor e to e with respect to its initial amplitude A .  When compressibility begins to play a role, the problem becomes more complex, because the most unstable waves are often oblique waves. As a consequence, a new parameter enters the dispersion relation: the angle t between the streamwise direction and the wavenumber vector. It is still assumed that the ampli"cation takes place in the x-direction only, i.e. b "0 or tK "0, but b G P (or t) needs to be speci"ed. To compute the local growth rate of a given frequency, the following procedure is often used: at each streamwise location, one seeks for the direction t of the wave which gives the maximum value + !a of a , and the N factor is de"ned by G+ G V !a dx N"Max G+ V D V "Max Max (!a ) dx. (11) G V R D This is the so-called envelope method.

 

4.1.2. Three-dimensional yows 4.1.2.1. Integration strategies. The extension of the e, method to three-dimensional #ows is not straightforward. The "rst reason is that the assumption b "0 is not G necessarily correct. Hence b must be assigned or comG puted. Several solutions have been proposed to solve this problem, see review in [26]. For instance, it is possible to use the wave packet theory and to impose the ratio *a/*b to be real [27]. A simpler solution is to assume that the growth direction is the group velocity direction or the potential #ow direction. In the case of in"nite swept wings, it is often assumed that there is no ampli"cation in the spanwise direction. After one of the previous assumptions for b has been G adopted, one has to integrate the local growth rates in order to compute the N factor. Several strategies are available: E Envelope method: this strategy was previously described for two-dimensional, compressible #ows.

E Fixed frequency/"xed spanwise wavenumber method ("xed u/"xed b method): the integrations are performed by following waves having a given dimensional frequency f and a constant value of the dimensional component b of the wavenumber vector in the spanP wise direction. The N factor is computed by maximizing the total ampli"cation with respect to both f and b . In other words, the N factor represents the envelP ope of several envelope curves. As for the envelope method, transition is assumed to occur for some more or less `universala value of N. In the following, this method will be referred to as the CSW (Constant Spanwise Wavenumber) strategy. This is the only strategy which is consistent with the PSE formulation. E Fixed frequency/"xed wavelength method ("xed u/"xed j method) and "xed frequency/"xed direction method ("xed u/"xed t method): as a wave of "xed frequency moves downstream, the wavelength or the propagation direction of the disturbances is kept constant. These strategies resemble the previous one in this sense that the N factor represents the envelope of several envelope curves. E Streamwise N factors/cross#ow N factors (N /N 21 !$ method): the principle is to compute a N factor for streamwise disturbances (the so-called N ) and an21 other N factor for cross#ow disturbances (the so-called N ). Transition is assumed to occur for particular !$ combinations of these parameters.

4.1.2.2. Examples of application. As it could be expected, each strategy gives a di!erent value of the N factor at the onset of transition. This problem has been studied quite recently in the framework of the ELFIN project funded by the European Community. The ELFIN project, ended in 1996, constituted a collaborative venture, bringing together the majority of European airframe manufacturers, research institutes and universities. In the "rst phase of the project, NLF (Natural Laminar Flow) experiments were conducted on a glove bonded to the original wing surface of a Fokker100 aircraft. This aircraft was instrumented with two infrared cameras for transition detection, one above and one below the wing. A series of #ight tests was conducted in 1991. In parallel, two series of HLFC experiments (Hybrid Laminar Flow Control) were carried out in the S1MA wind tunnel (Modane ONERA Centre) on a large model equipped with a suction system in the leading edge region. These experimental investigations provided a large amount of data which were then analysed within the second phase of the project. The main objective of the numerical work performed within ELFIN was to apply di!erent strategies of integration of the N factor in order to compare their capabilities and shortcomings. A summary of this work is given

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in [28,29]. The main conclusions are the following ones: E The `envelope-of-envelopesa methods ("xed u/"xed b, "xed u/"xed j, "xed u/"xed t strategies) provide quite similar N factors at the onset of transition; in other words, a wave travelling at a constant value of b keeps nearly constant values of j and t. The major shortcomings of these methods are that they are time consuming and that they produce many `pathologicala cases, i.e. N factor curves with a maximum located upstream of the measured transition point; an example of such behaviour is shown in the left-hand side of Fig. 6. E The N /N approach also produces `pathologicala !$ 21 cases. E The envelope method is the clear winner with regard to pathological cases: the N factor curves are usually monotonically increasing. Another advantage is that the computing time is low. E However, no method is superior for the purpose of N factor correlation at the transition point. Systematic comparisons between di!erent approaches of the e, method have been performed at ONERA Toulouse during the recent years. An experimental set-up was especially designed for this purpose [30,31]. The experiments were carried out in the F2 wind tunnel at Le Fauga-Mauzac ONERA Centre. The model was a swept wing with a chord (normal to the leading edge) of 0.7 m and a span of about 2.5 m. The pressure side was equipped with 7 independent suction chambers (in the chordwise direction) from 5 to 25% chord. Transition was detected by hot "lm sensors and by infrared thermography. A systematic variation of the suction velocity < and of the angle of attack a has been performed for U

181

wind tunnel speeds of 75 and 95 ms\ and for sweep angles u of 40 and 503. Variations of these parameters made it possible to generate di!erent transition mechanisms, from the pure CF type to the pure TS type. For a given con"guration, suction velocity was held constant in the di!erent suction chambers. Computations were "rst performed by using the local approach (without curvature terms). Two integration strategies were used: the envelope method and the "xed u/"xed b (CSW) strategy. In order to analyse the results, the evolution of the wavenumber direction t was systematically computed for the most ampli"ed frequency leading to transition. It turned out that the t direction computed with the CSW strategy was nearly constant from the neutral curve to the measured onset of transition, a result which is coherent with the ELFIN project conclusions. When the envelope method is considered, the most unstable t direction exhibits very large variations. Without suction, t typically decreases from about 853 close to the leading edge (CF disturbances) to about 03 at the transition location; in other words, the N factor in this case represents the cumulative e!ects of CF and TS disturbances. When suction is applied, the CF contribution is more or less eliminated, and the instability process is mainly governed by quasi-TS waves. From these observations the idea arose to consider the mean value of t, denoted as tM , for the waves leading to transition. tM is simply de"ned by



1 V2 tM " t dx (12) x !x 2  V x and x represent the critical and the transition abscis 2 sas, respectively. The values of the N factor at transition are plotted as function of tM in Figs 3 (envelope method) and 4

Fig. 3. Local N factors at transition (envelope method). From [30,31].

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Fig. 4. Local N factors at transition (CSW strategy). From [30,31].

(CSW strategy). The results obtained with the envelope strategy exhibit large values of N for tM between 40 and 803, i.e. for cases where the N factor is the sum of a CF contribution and of a TS contribution. When there is only one dominant instability mechanism (tM smaller than 403 and tM close to 903), the N factor at transition is smaller and around 8. On the other side, the N factors given by the CSW strategy take into account a single instability mechanism, which is either TS or CF (there is only one point at an intermediate value of tM ), and the scatter in the values of N is smaller. In the present investigation, the values of N for `purea TS and `purea CF transition processes are not very di!erent, but it cannot be claimed that this result is universal: a di!erent wind tunnel with a di!erent model could provide smaller N factors in the TS range and larger N factors in the CF range, or the contrary. In the range 803(tM (903, the general trend of the CSW strategy results is a decrease of N when the suction velocity increases. This could be due to disturbances generated by the suction holes, which could enhance the amplitude of the stationary vortices generated by the cross#ow instability. This seems to be con"rmed by nonlinear computations (see [32]) and by recent experiments performed at ONERA [33]. 4.1.3. Shortcomings of the local e, method As the e, method is based on linear stability only, receptivity and non-linear mechanisms are not taken into account. In addition the non-parallel e!ects are neglected in the local procedure; these e!ects will be discussed in Section 4.3. Several particular problems arise for three-dimensional #ows. The "rst one is to choose the `besta strategy of integration of the N factor. The envelope method is widely used, but its physical meaning is not clear in many cases, especially for swept wing #ows in the vicinity of the

point of minimum pressure: rapid variations in t can be + observed around this point when the dominant instability suddenly changes from the cross#ow to the streamwise type. The other strategies are often used in order to avoid these discontinuous (and probably unphysical) evolutions, but, as stated before, they often lead to `pathologicala situations. From a practical point of view, the most important issue is the value of the N factor at the onset of transition. Concerning two-dimensional #ows, it is now admitted that the e, method with NK7 to 10 can be applied to predict transition for a wide range of #ows if the background disturbance level is small enough. In threedimensional #ows, the results depend, of course, on the strategy which is chosen to compute the N factor, but, even with the same strategy, the results are not very clear. In fact, all the strategies produce a large scatter in the values of N at the onset of transition. A possible reason is that cross#ow and streamwise instabilities are not initiated by the same type of forced disturbances. For instance, the cross#ow disturbances are very sensitive to micron-sized roughness elements which have no e!ect on streamwise disturbances. Another reason is that there exists a multiplicity of nonlinear processes before the breakdown to turbulence, depending on the relative part of stationary and travelling unstable modes. The large chordwise extent of the non-linear region for CF dominated transitions also explains the di$culties to predict transition by using a linear theory. However, the e, method remains a very practical and e$cient tool, especially for parametric studies. For a given test model and for a given disturbance environment, it is often able to predict the variation of the transition location when changing a parameter which governs the stability properties of the mean #ow (pressure gradient, suction rate, etc.).

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4.2. Non-local approach By contrast with the local approach, it is no longer necessary to impose additional conditions for computing non-local N factors in three-dimensional #ows. This is due to the fact that b is real and constant; because b is real, the ampli"cation vector has only one component in the x direction; because b is constant, the N factor will be computed in a way which is similar to the CSW strategy described previously for the local theory. This does not mean that the non-local approach produces a unique N factor; as pointed out previously, di!erent disturbance quantities lead to di!erent growth rates, which in turn produce di!erent N factors. It is clear that the PSE equations should be solved in a coordinate system which is coherent with the assumptions made on the mathematical nature of b. For the case of an in"nite swept wing, the obvious choice is to consider x as the direction normal to the leading edge and z as the spanwise direction. In this coordinate system, the curvature terms take into account the surface curvature of the body. Because the curvature e!ects are of the same order of magnitude as the non-parallel e!ects, it is not coherent to include them into the local stability equations, but they have to be accounted for in non-local stability computations. To summarize, PSE results with curvature can be compared only with local results obtained without curvature by using the CSW strategy. 4.3. Comparison between local and non-local results The experimental database obtained at ONERA on the swept wing with suction (Section 4.1.2) has been used

183

for the purpose of comparison between local and nonlocal approaches [30,31]. The results demonstrated that non-parallel e!ects are strongly destabilizing in regions where CF disturbances are dominant, i.e. close to the leading edge. However, introducing curvature terms in the PSE equations reduces the di!erence between local and non-local N factors. Fig. 5 shows the non-local N factors at transition given by the non-local approach with curvature terms included. If these results are compared with those given by the local CSW strategy without curvature (Fig. 4), the di!erence is negligible for TS dominated cases, whilst non-local e!ects lead to an increase *NK2 for CF dominated cases. But the local and non-local values of tM are always close together, and the unstable frequency ranges are quite similar. The problem of comparison between local and nonlocal results was also addressed within the EUROTRANS (EUROpean program for TRANSition prediction) funded by the EC (1996}1999, [34]). The main objective of the project was to extend the theoretical work previously performed within ELFIN by trying to improve the transition prediction methods. In particular, non-local computations, both linear and non-linear, were performed for selected Fokker 100 and S1MA test cases. The non-local, linear computations with curvature terms included, revealed that the N factors based on di!erent quantities (maximum of streamwise and spanwise #uctuating velocities, maximum of mass #ow disturbances, disturbance energy) could be signi"cantly di!erent at the measured transition point. In some cases, the di!erence *N was up to 2 for mean values of N in the range 9}11. In general, the non-local theory (with curvature) gave higher values of N factors at transition point, compared to local theory (without curvature). However,

Fig. 5. Non-local N factors at transition. From [30,31].

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Fig. 6. Comparison between local (left) and non-local (right) N factors for a ELFIN test case (Fokker 100 experiments, rn417ou). The measured transition location is xK1 m.

the di!erence between both theories was of the same order as the di!erence due to the di!erent de"nitions of the N factor. As a consequence, the non-local approach did not reduce the scatter in the N factor for the considered series of 19 test cases. An interesting advantage of the non-local theory is its ability to eliminate some of the `pathologicala cases appearing when using the local theory, see Fig. 6. 4.4. Simplixed stability methods As the use of the e, method is often time consuming, the development of simpli"ed methods presents an unquestionable practical interest. The simplest solution is to apply analytical criteria expressing relationships between boundary layer integral parameters at the transition point [35]. Another possibility is to use simpli"ed stability methods, the complexity of which is intermediate between analytical criteria and exact stability computations. The general principle is to compute the disturbance growth rate from analytical relationships which have been established from exact stability computations. As soon as the growth rate is known, the N factor can be estimated in a classical way. Such methods have been developed at ONERA, "rst for two-dimensional, incompressible and transonic #ows [36]. The principle is as follows: for a prescribed value of the dimensionless frequency F"2pfl/; and for a given C mean velocity pro"le, the curve representing the evolution of the local growth rate p"!a as a function of G Rd (displacement thickness Reynolds number) is ap proximated by two half-parabolas, see Fig. 7. It was also observed that the variations of p , R , R and R with +   + F could be represented by simple, algebraic relationships, for instance, R "KF# +

(13)

Fig. 7. Notations for the simpli"ed method.

K, E and the other coe$cients appearing in the corresponding expressions for p , R , and R are then +   expressed as functions of some relevant mean #ow parameters. For low-speed #ows, exact stability results for self-similar basic #ows showed that E for viscous instability, a relevant parameter is the shape factor H; E for strongly in#ectional instability, the relevant parameters are the velocity ; and the shear stress P taken G G at the in#ection point altitude. In the framework of the ELFIN project (see Section 2.2), the simpli"ed method was extended to the modelling of pure cross#ow instability at low speed [37]. According to Squire's transformation, the problem is reduced to a series of two-dimensional problems. The mean velocity pro"les projected in directions close to the cross#ow direction are highly in#ectional, so that the modelling based on the values of ; and P can be used. Current G G investigations are devoted to the development of a general three-dimensional method applicable to CF as well to TS disturbances.

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Using the simpli"ed methods reduce the computing time by several orders of magnitude; it also allows automatic initializations for exact stability computations.

5. Non-linear approaches As explained above, the linear stability theory together with the e, method provides practical prediction for the transition location. A major shortcoming in this procedure is related to the fact that the non-linear processes are ignored. This can explain the loss of e$ciency of the e, method in many cases. The aim of the present section is "rst to present a recent approach (non-linear PSE) which quanti"es the weakly non-linear e!ects. We will then summarize the procedure for the application of this tool in a practical application. Some strong instabilities which are still missing in the weakly non-linear approach will also be described. 5.1. Weakly non-linear instabilities with the PSE approach

Fig. 8. Flat plate test case, subharmonic resonance.

With an appropriate initialization procedure, the set of coupled equations of the non-linear PSE approach (Section 3.4) is numerically integrated by a marching procedure in x similar to the one used in the linear approach, except that convergence is now required for each mode. Depending on the considered primary modes and on their initial amplitude, di!erent `scenariosa (hence di!erent types of results) are obtained. They can be divided roughly in three families. The "rst one is related to the two-dimensional mean #ow instability and to the TS instability on a swept wing. In this case, successive resonances for high harmonics occur up to a numerical divergence of the PSE code. The second family corresponds to the pure CF instability and exhibits usually a quasi equilibrium saturated state. The last family which is less documented concerns intermediate cases, for which TS and CF disturbances play simultaneously a role in the mechanism leading to the transition. This paragraph is devoted to a description of the "rst two scenarios.

initialized with two modes: a primary mode and a secondary mode. The primary mode is denoted as (2, 0); n"2 corresponds to the most dangerous frequency and m"0 is imposed because of the two-dimensional nature of the primary instability, see the mathematical form of the perturbation in Eq. (8). Two types of secondary mode can be introduced: the (2, 1) mode for the fundamental resonance (oblique mode with the same frequency as the primary mode) or the (1, 1) mode for the subharmonic resonance (half frequency). An example of subharmonic resonance is given in Fig. 8. This case has been studied by DNS at the University of Grenoble (IMG/LEGI), see [38]. The DNS results are represented by thick lines and are in good agreement with the non-linear PSE results (thin lines). The numerical results show "rst a linear evolution of the primary (2, 0) mode up to the streamwise location where its amplitude reaches a threshold. Then the secondary (1, 1) mode exhibits a rapid increase followed by other resonance mechanisms acting on higher harmonics and on the mean #ow distortion, the latter being characterized by the (0, 0) mode. This mode modi"es the mean boundary layer pro"le and may a!ect mean #ow quantities such as the shape factor or the skin friction coe$cient. Let us consider now the TS instability occurring on a swept wing. In contrast with the #at plate case, the secondary instability mechanism is not clearly de"ned in terms of the primitive space coordinates because the mean boundary layer is three dimensional and the primary TS wave is more or less aligned with the external streamline which is oblique with respect to the x direction. Let us consider again the free #ight test case described in Section 4.2. This case has been extensively studied

5.1.1. Two-dimensional and streamwise instabilities For this scenario, the laminar-turbulent transition on a swept wing is assumed to be dominated by TS waves. It has been already pointed out that this streamwise instability is more or less similar to the two-dimensional #ow instability. Therefore, it is useful to consider "rst the latter case, which has been extensively studied. The #at plate basic #ow is steady and two dimensional. Two successive instability mechanisms are observed: the #ow becomes "rst unsteady (primary instability) and then three dimensional (secondary instability, occurrence of the peak-valley structure). In order to model these two mechanisms, the non-linear PSE computation is

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Fig. 9. Example of non-linear PSE computation for a TS dominated transition on a swept wing.

within the European project EUROTRANS in the framework of the linear stability theory, see Fig. 6, and also in the framework of the non-linear PSE approach. The primary wave, i.e. the (1, 1) mode, corresponds to the most linearly ampli"ed mode at the experimental transition location (x /cK0.3). Its dimensionless initial 2 amplitude has been "xed to 10\, a value which is somewhat larger than the amplitude at the neutral curve, see Fig. 9. The chosen amplitude is the maximum rms value of the streamwise disturbance velocity component for each mode, normalized with the incoming #ow velocity. The secondary mode is the (0, 1) mode. The wave angles of the primary and of the secondary waves with respect to the external streamline are close to 403 and 853, respectively. For a low-speed #at plate #ow, the wave angle of the primary wave is 03; the present non-zero value comes from compressibility e!ects (the #ow is transonic) and also from the three-dimensional nature of the boundary layer. The secondary wave thus makes an angle of 453 with respect to the primary wave, a value which lies in the usual range for the #at plate case. The major di!erence is that the secondary wave for a swept wing is usually taken steady. This means that the resonance of the (1, 1) mode in Fig. 8 does not occur on a swept wing, whereas the resonance of the (0, 1) mode in Fig. 9 corresponds more or less to the resonance of the (0, 2) mode in Fig. 8. The results are plotted in Fig. 9. The symbols correspond to the linear evolution of the primary and secondary waves, whereas the curves represent the non-linear PSE simulation. Nine modes have been plotted. Three di!erent line thicknesses have been used, each one corresponding to a given frequency (very thick for n"0, thin for n"2, intermediate for n"1); the dashed lines corres-

pond to m"0, the full lines to m"1 and the dotted lines to m"2. The "rst result is the deviation from the linear prediction which occurs at x/cK0.21 simultaneously for the primary and secondary modes. The primary mode seems to saturate, whereas the secondary mode, which is nearly marginal according to the linear theory, exhibits a rapid and strong increase up to a large amplitude at the end of the computation (u (0, 1) close to 10% of the incoming

 velocity). The second interesting feature is the very rapid increase of the highest harmonics at the end of the computational domain (higher harmonics not represented here would exhibit a larger increase). For example, the (2, 0) mode grows by a factor 10 over a length less than 10% of the chord. At the end of the computation, the curves for di!erent modes with the same frequency become close together; the higher the frequency is, the smaller the "nal amplitude is. This behaviour is also observed for the #at plate instability mechanisms. It is also important to note that the steady modes, especially the (0, 1) and (0, 0) modes, reach a large amplitude. Their increase is similar to the growth of the (0, 0) and the (0, 2) modes obtained in Fig. 8. These modes can modify significantly mean #ow quantities, such as the skin friction coe$cient, as illustrated in Fig. 10. The line corresponds to the laminar boundary layer calculation. Its wavy evolution is due to the wiggles which are present in the measured C distribution. By the way, it can be noticed N that the non-linear PSE code is numerically robust, so that its application is not limited to idealized test cases with smooth C distributions. The squares in Fig. 10 N represent the skin friction coe$cient with the (0, 0) mode added to the mean boundary layer. The triangles

D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

Fig. 10. Skin friction evolution, non-linear e!ects of the steady modes.

represent the values which are obtained when the maximum contribution (with respect to z) of the secondary mode is also added. A deviation from the `pure laminara value is clearly observed, at x/cK0.3, close to the experimental transition location. This result can be interpreted as a transition criterion. To sum up, the weakly non-linear streamwise instability seems to be well understood; in addition a transition criterion may be deduced by looking at the increase of the skin friction coe$cient. Of course the corresponding abscissa is strongly dependent on the chosen primary mode and on its initial amplitude. This scenario has been investigated for the ATTAS #ight test cases by Schrauf [40]. 5.1.2. Crossyow instability In the case of cross#ow instability, the stationary modes are linearly ampli"ed. This implies that at least one stationary mode must be imposed at the "rst abscissa as a primary mode. In fact, two types of non-linear PSE computations are usually performed for CF dominated transitions: one with stationary modes only and one with a combination of stationary and unsteady modes. E In the "rst case, the (0, 1) mode is the only primary mode (as before, the value of the spanwise wavenumber is taken as the most dangerous according to the linear stability properties) and the computation provides the evolution of the di!erent (0, m) modes. This scenario has been successfully investigated within the AG27 GARTEUR group; a very good agreement between PSE results and DNS [41] results was achieved. E In the second case, in addition to the (0, 1) primary mode, an unsteady (1, 1) mode is usually introduced with the same spanwise wavenumber and with some `relevanta frequency. This mode is not a secondary mode; it can be considered as a second primary mode.

187

Fig. 11. Example of non-linear PSE computation for a pure cross#ow instability, comparison with DNS performed at IAG Stuttgart.

In this case, the problem of choosing a scenario arises: (i) the spanwise wavenumber may be chosen to be the most ampli"ed for the zero frequency mode, and then the frequency is chosen to be the most ampli"ed for this spanwise wavenumber; (ii) the frequency and the spanwise wavenumber of the travelling mode are those corresponding to the most ampli"ed (1, 1) mode and the (0, 1) mode is then automatically de"ned. Some compromises or intermediate choices may be also used. A typical result is given in Fig. 11 with two primary waves (i.e. with a (0, 1) mode and a (1, 1) mode). It is important to note that, by contrast with the TS case discussed before, the (1, 1) mode is now a cross#ow mode: its wave angle is not more or less aligned with the external streamline but makes an angle close to 803 with this streamline, as it is also the case for the (0, 1) mode. This example corresponds to experiments performed by Bippes at DLR GoK ttingen [6] and computed with a DNS at IAG (Institut fuK r Aerodynamik und Gasdynamik) in Stuttgart by MuK ller [42]. The experiments have been performed under controlled (i.e. forced) conditions. The values of the spanwise number and of the primary frequency used in the computations correspond to the forced ones. Four modes are represented in Fig. 11. The symbols give the linear evolution of the (1, 1) mode, the thick lines are the DNS results for the four modes and the thin lines are the PSE results. In this "gure, the maximum r.m.s. values of the streamwise disturbance velocity component for the di!erent modes are plotted as function of a non-dimensional distance from the leading edge, see [42] for details. The main result is that a non-linear saturation is observed: the amplitude of the (1, 1) mode remains more or less constant for x52.8, whereas the linear theory

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predicts a more or less constant growth. The corresponding amplitude is rather large (about 15% of the free stream velocity), in agreement with experimental results. The agreement between DNS and PSE is also very satisfactory. Comparing both approaches, it must be "rst remarked that the DNS provide reference results and are therefore absolutely necessary for the validation of the non-linear PSE approach. On the other hand, it may be noted that the CPU time for the PSE computation is much smaller than for DNS (1 h in comparison with 400 h, both on a CRAY) and that PSE computations are not limited to simple geometries, as it is still the case for DNS computations. However, even if the weakly non-linear mechanisms are satisfactorily modelled, neither the DNS nor the PSE computations exhibit any particular behaviour at the measured transition location (in the present case, transition occurs at xK3.5). As mentioned before, when transition is dominated by TS waves, high harmonics and steady modes exhibit a rapid resonance, with a resulting increase of the skin friction coe$cient which can be used to estimate the transition location. Nothing similar is observed in Fig. 11: non-linear interactions of pure CF disturbances lead to a quasi equilibrium state. This indicates that something is missing for transition prediction. This point is discussed in Section 5.3. 5.2. Practical use of non-linear PSE for swept wing yows The following procedure is currently used in order to apply the non-linear PSE approach in a practical case. First the mean #ow is assumed to be known, typically from a laminar boundary layer computation. Then a linear stability analysis is performed with the CSW strategy by solving either the local or the non-local equations. Then the critical parameters (frequency and spanwise wavenumber) are determined at the transition location. If this location is not known, it can be estimated, for example by using the e, method. The wavenumber angle of the dominant disturbances provides the nature of the instability which is responsible for transition (streamwise, cross#ow or intermediate). This leads now to the choice of the primary modes, i.e. the scenario of the non-linear instability process. 5.2.1. Scenarios For a pure streamwise instability, the (1, 1) mode being determined as explained above, another mode is introduced; this mode can be the (0, 1) mode, or the (1, 0) mode, or both (see [32] for di!erences which result from this choice). For a pure cross#ow instability, it is possible to restrict the analysis to steady modes. This scenario is relevant when the free stream turbulence level is very small (so that the initial amplitude of the travelling modes is assumed to be negligible), as it is the case for free #ight

conditions or in `quieta wind tunnels. If the environmental disturbance level is not so low, the classical solution consists of imposing a (1, 1) mode and the corresponding (0, 1) mode, see discussion in the previous paragraph. Intermediate cases have been poorly investigated. The di$culty is similar to that linked to the use of the envelope method in such cases: the streamwise instability starts in a boundary layer in which the cross#ow instability is still present, with a rather large value of the N factor. The most ampli"ed frequencies and spanwise wavenumbers are usually very di!erent for both kinds of disturbances. However, if one wants to apply the non-linear PSE approach, some proportionality between the integers n and m for the TS and CF disturbances has to be imposed. This leads to strange scenarios such as the interaction between a (0, 1) mode (CF) and a (3, 1) mode (TS), which has been investigated within the AG27 GARTEUR group. 5.2.2. Initial amplitudes In all the scenarios listed above, the initial amplitudes of the chosen primary modes are not known. They are usually determined by trial and error until an `interestinga (not trivial) non-linear behaviour is obtained. More precisely, the usual trends are the following ones. If the initial amplitudes are too small, the primary modes remain linear throughout the PSE computation. On the other hand, if the initial amplitudes are too large, a lot of modes exhibit a strong increase and the computation breaks down rapidly. The classical rule consists in searching initial amplitudes allowing numerical convergence up to the (known or estimated) transition location and leading to some deviation from a purely linear evolution of the primary modes. 5.3. High-frequency secondary instability Secondary instability means a loss of a symmetry property (which arises after a primary instability mechanism). In the example shown in Fig. 8, the growth of the (1, 1) mode implies that the #ow is no longer two dimensional. However another secondary instability is likely to occur in the laminar swept wing boundary layer just before the occurrence of the "rst turbulent spots. It is a highfrequency mechanism, which has been observed in several wind tunnel experiments, see [5,21] for example. It is typical of a pure cross#ow instability. Taking into account a new base #ow consisting of the laminar boundary layer #ow on which the saturated (0, 0) mode and possibly the (0, 1) mode are superimposed, it is possible to study its linear stability properties by using the Floquet theory, as initially proposed by Herbert [23] for the subharmonic or fundamental resonance in the #at plate case. In fact these steady modes, which have a large

D. Arnal, G. Casalis / Progress in Aerospace Sciences 36 (2000) 173}191

amplitude, modify the mean #ow, which may become in#ectional and therefore highly unstable. Clearly, this instability is the missing ingredient for three-dimensional boundary layer transition prediction. Attempts to model this phenomenon are described in [43,44,41]. Furthermore, it can be guessed that the high-frequency secondary instability may also occur in the case of the streamwise instability: the observed growth in the skin friction is the trace of the growth of the steady modes but it cannot explain the occurrence of a turbulent spot. Modelling these phenomena is the challenge for the future.

6. Conclusion The purpose of the present paper was to summarize the current state-of-the-art of the laminar}turbulent transition prediction for swept wing #ows. Di!erent methods have been presented: simple ones such as the database method or more sophisticated ones with the non-linear PSE approach. As mentioned above, the instability processes which lead to the "rst occurrence of a turbulent spot may be basically divided in three successive steps: 1 * receptivity, 2 * linear instability, 3 * non-linear instability. The third step may be in turn divided in two parts, the weakly non-linear region and the strongly non-linear region. For a #at plate #ow as well as for a swept wing #ow (at least in low speed and transonic conditions), important progresses have been made for 40 yr for step 2 and for 10 yr for the weakly non-linear aspects of step 3. Concerning step 2, the improvements include theoretical aspects as well as practical tools. The di!erent types of instability (TS, CF or mixed) occurring on a swept wing are now well understood and modelled. The destabilizing e!ects of the boundary layer growth and the stabilizing e!ects of the body curvature have been clearly quanti"ed. On the other hand, important e!orts have been directed to the development of practical methods. These methods are based today on the e, method, the N factor being evaluated directly with the linear stability theory or more rapidly with the so-called database method. From this survey of the state of the art, at least two questions arise: E Are theoretical improvements still feasible in the near future, particularly for step 1 and the strongly nonlinear instability in step 3? Secondary instability theory could be used for explaining the sudden appearance of high-frequency disturbances in step 3. It `remainsa to include it into the available theoretical chain: linear stability theory#non-linear PSE computations (#secondary instability). Of course, this new chain must be tested systematically by comparison with experiments or DNS. Moreover, many

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researches are currently devoted to the receptivity (step 1). They include detailed experiments which are carried out in order to isolate and to quantify the roughness e!ects, the noise e!ects and the turbulence e!ects on CF and TS waves. These investigations also include DNS computations which are now feasible for more and more complex problems, due to the constant development of the computers. The recent use of the adjoint methods may be a very simple tool to quantify some `smalla (linear) receptivity mechanisms. E Is it possible to transfer any new theoretical improvement into an improvement of the current practical tool? The situation may not be so favourable. For example, the recent improvements due to the nonlinear PSE computations explain why the e, method may fail, especially for CF dominated problems. But, up to now, this non-linear approach needs more or less the knowledge of the (experimental) transition location! It may be guessed that this will remain true until an improved receptivity theory has led to better estimates of the initial conditions. In any case, it seems clear that the non-linear PSE are too time consuming for practical purposes. Consequently a transfer is needed through a simpli"ed method, which contains this sophisticated physics. But is it meaningful to modify the linear N factor values by taking into account the `truea non-linear mechanisms? Or do we need a new method, completely di!erent from the hardwearing e,? Let us turn now to the industrial point of view. If we want to perform a #ight test on a laminar glove, what do we need? Are the sophisticated tools described above necessary? For a swept wing boundary layer, the answer may be negative: simple methods are often able to provide satisfactory predictions. But if we consider the case of a nacelle for example, di!erent stability mechanisms are likely to occur. Then the e, method calibrated from swept wing results may be less accurate. In this sense, it is necessary to develop general tools which could be applied to any boundary layer.

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