Non-parallel stability analysis of three-dimensional boundary layers along an infinite attachment line

Fluid Dynamics Research 27 (2000) 143–161

Non-parallel stability analysis of three-dimensional boundary layers along an in nite attachment line Nobutake Itoh ∗ National Aerospace Laboratory, 7-44-1 Jindaijihigashi-machi, Chofu, Tokyo 182-8522, Japan Received 26 April 1999; received in revised form 2 September 1999; accepted 17 December 1999

Abstract Instability of a non-parallel similar-boundary-layer ow to small and wavy disturbances is governed by partial dierential equations with respect to the non-dimensional vertical coordinate  and the local Reynolds number R1 based on chordwise velocity of external stream and a boundary-layer thickness. In the particular case of swept Hiemenz ow, the equations admit a series solution expanded in inverse powers of R21 and then are decomposed into an in nite sequence of ordinary dierential systems with the leading one posing an eigenvalue problem to determine the rst approximation to the complex dispersion relation. Numerical estimation of the series solution indicates a much lower critical Reynolds number of the so-called oblique-wave instability than the classical value Rc = 583 of the spanwise-traveling Tollmien–Schlichting instability. Extension of the formulation to general Falkner–Skan–Cooke boundary layers is proposed in the form of a double power series with respect to 1=R21 and a small parameter  denoting the dierence of the Falkner–Skan parameter c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All m from the attachment-line value m = 1. rights reserved.

1. Introduction Accurate theoretical prediction of transition location in three-dimensional boundary layers is increasingly required from aeronautical engineering, because technology of laminar- ow control is considered to be indispensable for improvement of fuel eciency in future development of high-speed aircraft. It is well known that instability and transition to turbulence of the boundary layer on most swept wings of modern aircraft occurs in the front region near the attachment line. However, theoretical studies of the attachment-line instability are rather limited and our present knowledge of the associated phenomena is not yet sucient for accurate prediction of transition location on a swept wing (see Reed and Saric, 1989; Reed et al., 1996, for instance). Since this ow eld is highly non-parallel, we may expect that more detailed stability analyses of the attachment-line boundary ∗

Fax: +81-422-40-3235. E-mail address: [email protected] (N. Itoh). c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/00/$20.00 All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 9 9 ) 0 0 0 4 7 - 7

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N. Itoh / Fluid Dynamics Research 27 (2000) 143–161

layer will give an important inspection into several fundamental eects of the boundarylayer non-parallelism. The most important contribution to this research eld may be the theoretical study by Hall et al. (1984), who considered plane-wave disturbances with the wavenumber vector parallel to the attachment line and derived an exact ordinary dierential system of linearized disturbance equations. (Extended studies were made by Lin and Malik (1996), Balakumar and Trivedi (1998) and others). If we consider oblique-wave disturbances with a chordwise component of the wavenumber vector, however, the linear disturbance equations have the coecients depending on two spatial coordinates in the chordwise and normal-to-wall directions and consequently remain in the original partial dierential form. On the other hand, multiple-instability characteristics of the attachment-line boundary layer were revealed in a recent study (Itoh, 1996b), where eigensolutions were obtained by solving a simple model of stability equations consisting of the parallel- ow approximation plus a few non-parallel terms. Important ndings are that this ow is susceptible to two kinds of instabilities, namely the cross- ow instability inherent in three-dimensional boundary layers and the streamline-curvature instability recently discovered by the present author (Itoh, 1994b), both of which yield oblique-wave disturbances, and that the critical Reynolds number to such oblique waves is much lower than the critical value Rc = 583 to spanwise-traveling disturbances obtained by Hall et al. (1984). This situation brings out new questions as to the relationship and the selection mechanism between the spanwise-traveling disturbances and the oblique-wave disturbances in more realistic ow conditions, say, in the front region of a swept wing. It should be noted here that the attachment-line ow (swept Hiemenz ow) belongs to Falkner– Skan–Cooke similarity solutions of the three-dimensional boundary-layer equations. Such a similar boundary layer has a great mathematical advantage that the non-dimensional coordinate in the chordwise direction perpendicular to the leading edge is identical with the chordwise Reynolds number R1 de ned by the chordwise velocity component of external stream and a local boundary-layer thickness. Then the exact equations governing oblique-wave disturbances are written in a partial dierential form with respect to the non-dimensional normal-to-wall coordinate and the chordwise Reynolds number. An important study relevant to this fact was recently made by the present author (Itoh, 1998), who considered linearized disturbance equations for Gortler instability of two-dimensional similar boundary layers along a concave wall and proposed an expansion method to seek a double power-series solution of the exact partial dierential equations. In the particular case of stagnation-point ow, this solution is reduced to a single series with respect to inverse powers of square of the Reynolds number R1 . It may be emphasized that the leading terms of the series solution are determined from an eigenvalue problem of the lowest-order ordinary dierential system. As a matter of fact, a similar method of expansion can be applied to the stability problem of three-dimensional boundary layers, if the basic laminar ow is given by a member of the Falkner–Skan–Cooke similarity family, including the attachment-line ow. In this problem, the spanwise Reynolds number based on spanwise velocity component of external stream appears as a stability index independent of the expansion parameter R1 instead of Gortler number. In the present study, we rst derive the exact partial dierential equations governing small oblique-wave disturbances superimposed on a steady boundary-layer ow with one of the Falkner– Skan–Cooke velocity pro les developing on an in nitely long yawed wedge. In general cases, the non-dimensional frequency !, the spanwise wavenumber ÿ and the spanwise Reynolds number R0 are functions of the chordwise Reynolds number R1 , which acts as the chordwise coordinate, but for

N. Itoh / Fluid Dynamics Research 27 (2000) 143–161

145

Fig. 1. Flow eld and coordinates.

the particular case of swept Hiemenz ow, the boundary-layer thickness is independent of R1 , so that the above quantites !; ÿ and R0 become constant. The disturbance equations for this particular basic ow admit a series solution expanded into inverse powers of R21 and are then decomposed into an in nite sequence of ordinary dierential equations with the leading equation system posing an eigenvalue problem to determine the rst approximation to the chordwise complex wavenumber  and the corresponding velocities and pressure of oblique-wave disturbances. Next, an approximate method of expansion is proposed to derive a series solution of the disturbance equations for general

ows in the Falkner–Skan–Cooke family. In this case, the solution is represented by a double power series with respect to 1=R21 and a small parameter  denoting a dierence of the basic ow from the attachment-line pro le, and the leading terms of the series solution are again determined from an eigenvalue problem of the lowest-order equation system. Finally, numerical investigations of those series solutions are made to reveal essential eects of a nite value of the Reynolds number R1 on the oblique-wave instability of the attachment-line ow and to estimate relative magnitudes of the second and third terms in the  expansion of the series solution for general Falkner–Skan–Cooke boundary-layer ows. This will give an index for validity of eigensolutions of the lowest-order equation system. 2. Governing equations This study investigates linear stability of three-dimensional boundary layers on a yawed wedge placed in a uniform ow. We use a rectangular coordinate system with x denoting the chordwise distance in the direction perpendicular to the leading edge, y the spanwise distance along the leading edge and z the normal distance from the surface, as schematically shown in Fig. 1, and let t denote time, (U + u∗ ; V + v∗ ; W + w∗ ) velocity components, P + p∗ pressure,  density and  kinematic viscosity, where the quantities u∗ ; v∗ ; w∗ and p∗ represent small disturbances superimposed on  It is well known that the laminar ow on a yawed wedge the laminar basic ow (U ; V ; W ; P). is described by the Falkner–Skan–Cooke similarity solutions of the boundary-layer equations. Each

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member of this family is subjected to an external ow with the chordwise velocity UE proportional to xm and the spanwise velocity of a constant value V∞ , where m is a constant parameter associated with the angle  of the wedge as  = 2m=(1 + m) and indicates the magnitude of the pressure gradient (see, for p instance, Rosenhead, 1963, Chapter 8). Introducing the boundary-layer thickness de ned by  = x=UE and the dimensionless variable  = z=, we can write the similarity solution in the form   1−m 0 1+m  0    F() − F () ; U = UE (x)F (); V = V∞ G(); W =− (2.1) (x) 2 2 where the prime denotes dierentiation with respect to  and the functions F and G are obtained from numerical solution of the ordinary dierential system 1 + m 00 1+m 0 FF + m{1 − (F 0 )2 } = 0; FG = 0; F 000 + G 00 + 2 2 F(0) = F 0 (0) = G(0) = 0;

F 0 (∞) = G(∞) = 1:

(2.2)

This velocity family includes the Blasius ow on a at plate at m = 0 and the attachment-line ow (swept Hiemenz ow) on a plane wall at m = 1 and is very important in the practical sense that the member can be eectively used as an approximation to a local velocity distribution of the boundary layer on a slender swept wing with a pressure gradient only in the direction perpendicular to the leading edge (Itoh, 1996a). In this section, we derive the exact stability equations which govern the small disturbance (u∗ ; v∗ ; w∗ ; p∗ ) superimposed on the above basic ow. It is usual in linear stability calculations to choose either temporal or spatial approach; temporal theory assumes frequency of disturbances to be complex with keeping wavenumbers real, while spatial approach assumes the wavenumber in the chordwise direction to be complex instead of the frequency. In the present study, one of the main purposes is to investigate non-parallelism of boundary-layer ows, which will appear in the disturbance equations through the x-dependence of various quantities, such as the chordwise wavenumber and ampli cation rate. For a rational expression of such variations, therefore, we take the spatial approach here and assume the disturbance velocities and pressure to be written in the form ∗

∗

∗

 Z

∗

(u ; v ; w ; p ) = (u;  v;  w;  p)  exp i

 − !t  d x + ÿy 



;

(2.3)

where the frequency ! and the spanwise wavenumber ÿ are constant but the chordwise wavenumber  is complex and a function of x to be determined later, and the amplitude functions u;  v;  w and p may be considered to depend on two space variables x and z only, because the basic ow is stationary and independent of the spanwise coordinate y. If we substitute these expressions into the Navier–Stokes and continuity equations, subtract the basic ow parts, which are assumed to satisfy the equations by themselves, and linearize with respect to the small disturbance, then exact disturbance equations are obtained in the form "



#





  @ @U @ @U 1   2 u;   + iÿ V + W u + i + p + ∇  + w = − @z @x @z  @x





 @ @V @V p   2 v;   + iÿ V + W v +  u + w = −iÿ + ∇ @z @x @z 

@ −i! + U i + @x

@ −i! + U i + @x

N. Itoh / Fluid Dynamics Research 27 (2000) 143–161

"



@ −i! + U i + @x





147

#

@W 1 @p @ @W  2 w; + u = − + ∇ + iÿ V + W w +  @z @z @x  @z



@ @w = 0; i + u + iÿ v + @x @z

(2.4)

2

2

 = (i + @=@x)2 − ÿ + @2 =@z 2 , and overbars indicate dimensional quantities. Since the basic where ∇

ow varies slowly in the x direction, we may consider that solutions of the above partial dierential equations will only weakly depend on the chordwise coordinate in comparison with the dependence on the vertical coordinate. An appropriate nondimensionalization is necessary for a clearer and more rational exhibition of this dierence in spatial variations. Expression (2.1) for the basic laminar ow implies that the reference p length suitable for the present stability analysis is the boundary-layer thickness de ned by  = x=UE , but that there are two velocity scales UE and V∞ denoting the chordwise and spanwise velocity components, respectively, of local external stream. This naturally leads to two Reynolds numbers de ned by R0 =

V∞  ; 

R1 =

UE  ; 

(2.5)

the latter of which depends on x through both UE and  and so is proportional to x(1+m)=2 , while the x dependence of R0 is of the form x(1−m)=2 coming from  only. We also use the reference length and velocities to non-dimensionalize all quantities appearing in the disturbance equations (2.4) as x = R1 ; 

z = ; 

U = U (); UE

 = ;

V = V (); V∞ v = v(; R1 ); V∞

u = u(; R1 ); UE

ÿ = ÿ;

! = !; V∞

W W () = ; V∞ R0 W = w(; R1 ); V∞

p = p(; R1 ); 2 V∞

(2.6)

where the chordwise velocities U and u have been scaled with the local velocity UE but not with the constant velocity V∞ , because this scaling is used in deriving rigorous stability equations for spanwise-traveling disturbances, as shown by Hall et al. (1984). Of particular importance here is that the boundary-layer thickness de ned in this paper causes the dimensionless coordinate x= in the chordwise direction to be identical with the chordwise Reynolds number R1 given in Eq. (2.5). It will be shown later that this identi cation is a great advantage of similar boundary-layer ows in the simpli cation of stability equations. Now substitution of Eq. (2.6) into Eq. (2.4) leads to the exact disturbance equations in the dimensionless form     1 ˆ2 @ R1 1 − m 1+m W 2m 1−m 0 D + R1 − iÿV − D − U+ U u ∇ + i! − U i − R0 R0 2R0 2R0 @R1 R0 R0 2R0 



@ R2 R1 1 − m 1+m D + R1 p = 0; − U w − 02 i − R1 R0 2R0 2R0 @R1 0

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1 2 @ R1 1 − m 1+m ∇ + i! − U i − D + R1 R0 R0 2R0 2R0 @R1 +





− iÿV −

W D v R0

1−m 0 V u − V 0 w − iÿp = 0; 2R0 

1 2 @ R1 1 − m 1+m ∇ + i! − U i − D + R1 R0 R0 2R0 2R0 @R1 +







W W0 − iÿV − D − w R0 R0

1−m (W + W 0 )u − Dp = 0; 2R20





@ R1 m 1−m 1+m i + − D + R1 u + iÿv + Dw = 0; R0 R0 2R0 2R0 @R1 where D ≡ @=@ and 



(2.7)



1 + m d 1−m 1−m 1+m @ −  + 2i − D + ∇ ≡D − −ÿ +i 2 dR1 2R1 2R1 2 @R1 2

2

2

1−m + 2R21

2





3−m 1 − m2 @ 1−m 2 2 (1 + 2D) + D + D − 2 2 4R1 @R1 







1+m 2

2



@2 ; @R21

2 1 + m d 1 − 5m 1−m 1+m @ ∇ˆ ≡ D2 − 2 − ÿ2 + i −  + 2i − D + 2 dR1 2R1 2R1 2 @R1



+



1−m 3 − 5m 1+m 1−m 2 2 D − −2m + D + 2R21 2 2 2R1

@ × + @R1



1+m 2

2





1 − 5m + (1 − m)D 2



@2 : @R21

These are partial dierential equations with respect to the two independent variables  and R1 and the quantities ; ÿ; ! and R0 included in the coecients are functions of R1 in general. Thus, the problem in the present study is to solve the above partial dierential equations subject to the usual boundary conditions that the disturbance velocity should vanish on the wall and far away from the wall; that is, u=v=w=0

at  = 0

and as  → ∞:

(2.8)

Following the previous study on Gortler instability (Itoh, 1998), we rst discuss a particular case of the attachment-line ow with m = 1 in the next section and then consider more general cases of Falkner–Skan–Cooke ows with m 6= 1 in Section 4. 3. Series solution for swept Hiemenz ow In this section, we consider the particular case where the basic laminar ow is given by the attachment-line boundary layer with the Falkner–Skan pressure parameter m = 1. This value of m

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149

extinguishes several terms proportional to 1 − m in the disturbance equations (2.7) and, in addition, yields a constant value of the boundary-layer thickness  and consequently, constant values of the frequency !, the spanwise wavenumber ÿ and the spanwise Reynolds number R0 which are non-dimensionalized with the reference length  and the constant reference velocity V∞ . We wish to have such a solution of the exact disturbance equations that provides a relation governing local stability characteristics of the ow; that is, an eigenrelation connecting the above three quantities !; ÿ; R0 , the chordwise wavenumber  and the chordwise Reynolds number R1 . Since dependence of  on the chordwise distance x is not yet speci ed, it may be plausible to assume that the disturbance velocities and pressure u(); v(); w(); p() and the complex wavenumber  are determined as functions of !; ÿ; R0 and R1 . Noting that the disturbance equations include the chordwise wavenumber and Reynolds number in the form R1 =R0 and 1=Rn1 , respectively, and that the dierential operator R1 (@=@R1 ) applied to 1=Rn1 yields only a constant multiplier −n, we seek a series solution of Eq. (2.7) in the form 



R0 1 1 = 0 (ÿ; !; R0 ) + 1 (ÿ; !; R0 ) 2 + 2 (ÿ; !; R0 ) 4 + · · · ; R1 R1 R1 u = u0 (; ÿ; !; R0 ) + u1 (; ÿ; !; R0 )

1 1 + u2 (; ÿ; !; R0 ) 4 + · · · ; 2 R1 R1

(3.1)

together with similar expressions for v; w and p. Now we substitute Eq. (3.1) into Eq. (2.7) and separate out each power of 1=R21 to obtain an in nite sequence of simultaneous equations for coecients of series (3.1). Due to the particular choice of m = 1, those equations are simpli ed to the ordirary dierential form 







1 2 2n − 2 (D − ÿ2 ) + i! − U i0 − R0 R0 

1 2 2n (D − ÿ2 ) + i! − U i0 − R0 R0 

1 2 2n (D − ÿ2 ) + i! − U i0 − R0 R0









− iÿV − 



W − iÿV − D un − U 0 wn = fn(1) ; R0 W D vn − V 0 wn − iÿpn = fn(2) ; R0 

W W0 − iÿV − D − wn − Dpn = fn(3) ; R0 R0



2n − 1 i0 − un + iÿvn + Dwn = fn(4) ; R0

(3.2)

where n=0; 1; 2; : : : . For the rst set with n=0, the forcing terms fn(1) ; : : : ; fn(4) are all zero, indicating that the lowest-order equation system is of homogeneous type. On the other hand, the forcing terms for n¿1 are given by    k−1 n  X  X  R0 fn(1) = k−j−1 j + i(4n − 2k − 3)k−1 + ik U un−k + ik−1 R20 pn−k    k=1

−

j=0

2 (n − 1)(2n − 3)un−1 − 2(n − 1)R0 pn−1 ; R0

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fn(2) =

fn(3) =

 n  X

R0

k=1



 n  X

fn(4) = −



n X

k−j−1 j + i(4n − 2k − 1)k−1 + ik U

j=0

R0

k=1

k−1 X

k−1 X

k−j−1 j + i(4n − 2k − 1)k−1 + ik U

j=0

     

vn−k −

2 (n − 1)(2n − 1)vn−1 ; R0

wn−k −

2 (n − 1)(2n − 1)wn−1 ; R0

ik un−k :

(3.3)

k=1

The lowest-order equation system of Eq. (3.2) is homogeneous and, together with the homogeneous boundary conditions (2.8), poses an eigenvalue problem to determine the leading terms of Eq. (3.1) as functions of !; ÿ and R0 . For comparison with conventional stability formulations for three-dimensional ows, we eliminate the components v0 and p0 from the above equations for n = 0 to obtain the simultaneous equations for u0 and w0 in the form 



1 2 W 2 (D − ÿ2 ) + i(! − 0 U − ÿV ) − D − U u0 − U 0 w0 = 0; R0 R0 R0



1 2 W W0 (D − ÿ2 ) + i(! − 0 U − ÿV ) − D − R0 R0 R0 





(D2 − ÿ2 ) + i(0 U 00 + ÿV 00 )



2 1 1 i0 + (UD + U 0 )u0 = 0: + (U 0 D + U 00 ) w0 + R0 R0 R0

(3.4)

If we put 0 = 0, these equations are reduced to the exact stability equations derived by Hall et al. (1984) for spanwise-traveling disturbances, while neglect of the U and W terms yields the Orr–Sommerfeld equation for spanwise parallel ows. In addition, the above form of equations is convenient for derivation of the matching boundary conditions imposed at an outer edge of the boundary layer. Using the usual procedure to derive the matching conditions (Itoh, 1994a), we have the approximate boundary conditions u0 = w0 = w00 = 0

at  = 0;

u00 − 1 u0 = w000 − (2 + 3 )w00 + 2 3 w0 + C1 u0 = w0000 − (1 + 2 + 3 )w000 + (1 2 + 2 3 + 3 1 )w00 −1 2 3 w0 = 0

at  = e ;

(3.5)

where 1 = We =2 − {We2 =4 + ÿ2 − iR0 (! − 0 − ÿ) + 2}1=2 ; 3 = We =2 − {We2 =4 + ÿ2 − iR0 (! − 0 − ÿ) + We0 }1=2 ; C1 = 2(1 + i0 R0 )1 ={R0 (1 + 3 − We )(1 + 2 )};

2 = −ÿ;

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151

and the subscript e indicates the value at  = e . The outer edge of the boundary layer is taken at a suciently large distance e = 10 in the computations given here, because numerical eect of a larger value, say e = 14, on eigenvalues is less than 0.01%, as fully discussed by Itoh (1994a). Once the eigenvalue problem is solved for given values of !; ÿ, and R0 , we can proceed to the calculation of the second-order terms in Eq. (3.1) by solving the inhomogeneous equations (3.2) for n = 1. At this stage, however, solution (3.1) is found to have an arbitrariness in magnitude of the amplitude, because the original equations (2.7) are linear and homogeneous. In order to remove this arbitrariness and to de ne a unique solution, it is necessary to impose, on each coecient of the series solution (3.1), an appropriate normalization condition, which must be consistent with the non-dimensionalization given by Eq. (2.6) and with the boundary conditions (2.8). In the present study, we take simple but de nite conditions of the form u00 (0) = 1;

u10 (0) = u20 (0) = · · · = 0;

(3.6)

where the prime denotes dierentiation with respect to . When the eigenfunction is normalized subject to (3.6), the forcing terms given by (3:3) for n = 1 are known functions except for 1 , which can be determined subject to the corresponding condition in Eq. (3.6) in the process of solving the inhomogeneous equation system. Here the boundary conditions to be imposed may be written approximately in the same homogeneous form as Eq. (3.5) with neglect of the forcing terms, because amplitudes of the eigensolutions are negligibly small at such a large distance from the wall as e =10, as can be seen in Fig. 5 given later. A similar procedure may be applied to the equation system (3.2) for n = 2 and so on to obtain higher-order terms in the series solution (3.1). 4. Double-expansion method for Falkner–Skan–Cooke boundary layers The formulation given in the previous section is concerned with the particular case of m = 1, including the constant parameters ÿ; ! and R0 , and has shown that the leading terms of the series solution (3.1) are obtained by solving an eigenvalue problem of the ordinary dierential equations, which form the lowest-order approximation to the exact disturbance equations (2.7). In this section, an attempt is made to extend the formulation to a more general form applicable to Falkner–Skan– Cooke boundary layers with m 6= 1, where ÿ; ! and R0 are not constant but depend on the chordwise coordinate R1 through the reference length  used in Eq. (2.6). If we assume that the solution is represented by a function of ÿ; !; R0 and R1 as given in the previous section, then the dierentiation with respect to R1 in the dimensionless equations (2.7) has to be rewritten as   1+m @ @ @ @ 1+m @ R1 R1 +! + R0 → + ÿ ; (4.1) 2 @R1 2 @R1 @ÿ @! @R0 where  stands for (1 − m)=2 and indicates chordwise variation of the boundary-layer thickness . This form of partial dierentiations suggests further expansion of solution (3.1) into power series of the parameter , which is small if we consider basic ows suciently near the attachment line on a swept wing. The parameter m is an index of velocity distributions belonging to the Falkner–Skan–Cooke similarity solution (2.1) and we wish to have such an eigenvalue problem that describes stability characteristics of each boundary-layer ow for a given value of m. This purpose is, however, not accomplished by the direct expansion of Eq. (3.1) with respect to the small parameter , because 

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N. Itoh / Fluid Dynamics Research 27 (2000) 143–161

is related to m by the de nition given above and therefore the formal expansion results in the same lowest-order equations as Eq. (3.4) for the attachment-line ow with m = 1. To meet these contradictory requirements, we rst introduce a false problem, which is de ned by the same equations as Eq. (2.7) after substitution of (4.1) but in which the small parameter  is assumed to be independent of the Falkner–Skan pressure parameter m. This false problem is then solved by expanding the solution into double power series of 1=R21 and , whose coecients are governed by an in nite sequence of ordinary dierential equations. After establishing the coecients, we may substitute  = (1 − m)=2 into the series to have a solution of the original equations (2.7). This false-expansion method gives reliable solutions at least for m suciently near unity and has a great advantage that the lowest-order eigenvalue problem will provide the rst approximation to stability characteristics of the given boundary-layer ow. The formulation begins with the equations governing the leading terms in Eq. (3.1), which may be written in the series form 0 = 00 + 01  + 02 2 + · · · + 0k k + · · · ; u0 = u00 + u01  + u02 2 + · · · + u0k k + · · · ;

(4.2)

together with similar expression for v0 , w0 and p0 . Before writing down the ordinary dierential equations, we introduce the quantities de ned by 1−m U; Wˆ = W − 2

wˆ 0k = w0k −

1−m u0k ; 2R0

(4.3)

for k = 0; 1; 2; : : : and use these instead of W and w0k hereafter, because the lowest-order equations are then written in a simpler form with a similarity to Eq. (3.2) for n = 0; that is, the modi ed vertical velocity of the basic ow is given by Wˆ = −(1 + m)F()=2 and coincident with the original form W used in Eq. (3.2) at m = 1. If Eqs. (4.1) and (4.2) are substituted into Eq. (2.7) and the transformation (4.3) is made, then the partial dierential equations are decomposed into a sequence of ordinary dierential equations of the form "

#

1 2 Wˆ 2m (1) (D − ÿ2 ) + i(! − 00 U − ÿV ) − D − U u0k − U 0 wˆ 0k = g0k ; R0 R0 R0

"

"

#

1 2 Wˆ (2) (D − ÿ2 ) + i(! − 00 U − ÿV ) − D v0k − V wˆ 0k − iÿp0k = g0k ; R0 R0 #

0 1 2 Wˆ 1−m Wˆ (D − ÿ2 ) + i(! − 00 U − ÿV ) − D − U− wˆ 0k R0 R0 2R0 R0



+



1−m 1+m (3) U u0k − Dp0k = g0k D+ ; 2 R0 2

(4) ; [(i00 + (1 + m)=2R0 ]u0k + iÿv0k + Dwˆ 0k = g0k

(4.4)

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153

where the forcing terms on the right-hand side are all zero for k = 0, while the forcing terms for k¿1 are given by (1) g0k =













k X @ @ U @ +! + R0 ÿ u0(k−1) + iU 0j u0(k−j) ; R0 @ÿ @! @R0 j=1

(2) g0k

k X @ @ U @ +! + R0 = ÿ v0(k−1) + iU 0j v0(k−j) ; R0 @ÿ @! @R0 j=1

(3) g0k

k X @ @ U @ +! + R0 = ÿ wˆ 0(k−1) + iU 0j wˆ 0(k−j) ; R0 @ÿ @! @R0 j=1

(4) g0k =−





k X @ 1 @ @ ÿ u0(k−1) − i 0j u0(k−j) : + R0 +! R0 @ÿ @! @R0 j=1

(4.5)

The rst homogeneous set for k =0 in the above equations poses an eigenvalue problem to determine the eigenvalue 00 and the corresponding eigenfunctions u00 (); v00 (); w00 () and p00 () as functions of !; ÿ and R0 . For k¿1, however, the equations are inhomogeneous with the forcing terms including the unknown parameter 0k , which is determined by the well-known solvability condition. Also, the !; ÿ and R0 derivatives of u0k ; : : : ; p0k appearing in the forcing terms can be obtained from dierentiating both sides of Eq. (4.4) with respect to !; ÿ and R0 , respectively, and solving the resulting inhomogeneous equations subject to the same solvability condition. The above procedure may be applied to higher-order terms in the series solution (3.1) to determine 1k ; u1k ; : : : ; p1k for k =0; 1; 2; 3; : : : and so on. Then we have a solution of double power series in 1=R21 and  for the case of general Falkner–Skan–Cooke basic ows. In this solution, the rst expansion parameter 1=R21 can take any small value without restriction, and therefore the leading terms of the series are meaningful in the asymptotic sense that R1 tends toward in nity. On the other hand, the second parameter  is not free but depends on the Falkner–Skan parameter m, so that the lowest-order solution obtained from the eigenvalue problem will be meaningful only if the series solution converges at the value =(1−m)=2 for a given value of m. Since it is very dicult to prove convergence or divergence of series (4.2) in general, we must be content with rather crude judgement from numerical computations. Such discussion will be given later together with the numerical results of the rst few coecients of Eq. (4.2). 5. Results of numerical computations The homogeneous and inhomogeneous equations of ordinary dierential form presented in the previous sections have been solved with a very accurate numerical method (see Itoh, 1994a) to show several fundamental properties of the lowest-order eigenvalue problem and practically important contributions of the higher-order correction terms in the series solutions (3.1) and (4.2). At rst, we consider instability of swept Hiemenz ow (the attachment-line ow with m = 1) to wavy disturbances traveling in the spanwise direction, which are exactly governed by the homogeneous

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Fig. 2. The neutral curve of the attachment-line ow to spanwise-traveling disturbances with 0 = 0 (solid line: neutral stability; chain-dotted line: maximum-growth-rate curve).

equations obtained by putting 0 = 0 in Eq. (3.4), because this form of disturbances allows the solution to be independent of the chordwise coordinate R1 . In this particular case with 0 = 0, it is usual to take temporal stability approach, where ! is complex and the positive and negative values of the imaginary part !i show the instability and stability, respectively, of the ow. Then we have the neutral stability curve given in Fig. 2, which is in complete agreement with the result of Hall et al., (1984) and indicates suciently high accuracy of the present numerical computations. If the disturbances are of oblique-wave type with non-zero values of the chordwise wavenumber 0 , however, the eigenvalue problem posed by Eqs. (3.4) and (3.5) yields positive ampli cation even at much lower values of the Reynolds number R0 and we can draw neutral curves on the (0 ; ÿ) plane for xed values of R0 , as shown in Fig. 3. Important ndings are that shape of the neutral curve becomes more complicated with the increase in R0 , suggesting superposition of dierent modes of instability, and that each neutral curve includes at least two open circles between which stationary disturbances with !=0 have positive growth rate. It should be noted that such stationary disturbances are dominantly observed in many experiments on three-dimensional boundary layers (see Radeztsky et al., 1994, for instance). Of more importance in this gure is that the oblique wave disturbances have the critical Reynolds number R0 = 87:1, which is very much lower than the critical value R0 = 583 for spanwise-traveling disturbances with 0 = 0 rst predicted by Hall et al. (1984). Fig. 4 shows the frequency !, the chordwise wavenumber 0r , the spanwise wavenumber ÿ and the spatial ampli cation rate −0i of the most ampli ed disturbance at each location of the Reynolds number R0 . It is very interesting to nd a sharp variation of the disturbance properties around R0 = 110 in this gure, because we can deduce from it the presence of two kinds of oblique-wave disturbances in the attachment-line ow. In fact, this deduction is con rmed by comparison of amplitude and phase distributions in the  direction of such eigensolutions as shown in Fig. 5, where the chordwise and spanwise velocities obtained at R0 = 87:1 and 200 on the maximum-growth-rate curves are compared with each other and also with those of the spanwise-traveling disturbance at the critical point R0 = 583. The chordwise velocity distribution u0 () of the eigensolution at R0 = 200 ◦ shows a phase shift of about 110 between  = 0 and 3, indicating the typical characteristics of the

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Fig. 3. Neutral stability curves of the attachment-line ow to oblique-wave disturbances in the (0 ; ÿ) plane for dierent values of R0 (solid circle: critical point; open circle: stationary disturbance with ! = 0).

Fig. 4. The maximum-growth-rate curves for oblique-wave disturbances in the attachment-line ow.

cross- ow mode, while a very small phase shift in u0 () of the eigensolution at R0 = 87:1 is inherent in the streamline-curvature mode. This dierence in phase distribution between the two instability modes is the most important key to distinguish one from the other in experimental observations of three-dimensional boundary-layer instability, as early pointed out by Itoh (1994b). It is also found in Fig. 5(b) that the spanwise velocity distribution v0 () of the eigensolution at R0 = 583 exhibits a ◦ sharp phase shift of 180 at  = 2 in contrast to rather gentle phase shifts of the two oblique waves. This sharp phase shift is the famous characteristic of the Tollmien–Schlichting viscous mode. From the maximum-growth-rate curve given in Fig. 4, we have the critical Reynolds number R0 = 87:1 of the attachment-line swept Hiemenz ow to oblique-wave disturbances. This critical value is, however, obtained only from the leading term of the series solution (3.1) and means the

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Fig. 5. Comparison of the eigenfunctions on the maximum-growth-rate curves in Fig. 4 and on the neutral curve in Fig. 2 (solid line: the oblique-wave disturbance at R0 = 200; dashed line: the oblique-wave disturbance at R0 = 87:1; dotted line: the spanwise-traveling disturbance at R0 = 583). (a) The chordwise velocity component u0 (). (b) The spanwise velocity component v0 ().

asymptotic value at the limit of R1 in nity. For nite values of R1 , therefore, we must take account of assigned contributions from higher-order terms of Eq. (3.1) by solving the inhomogeneous equation system (3.2) and then have the critical Reynolds number R0 varying with the chordwise location R1 . If the rst few coecients of Eq. (3.1) are calculated along the maximum-growth-rate curves given in Fig. 4, the spatial growth rate −i is determined as a function of R0 ; R1 and the truncation number N as   R0 1 N − i = − Im 0 + 2 + · · · + 2N ; (5.1) R1 R1 R1 and the condition i = 0 gives a simple approximation to the critical Reynolds number, as shown in Fig. 6. This gure indicates that the series solution may be considered to converge if R1 is larger than about 250 and that the critical value slightly increases from R0 = 87:1 q at R1 = ∞ to R0 = 125 at R1 = 250. If we introduce the local Reynolds number de ned by R ≡ R20 + R21 and the inclination angle  ≡ R1 =R0 of external stream from the spanwise direction, then the results given in Fig. 6 are transformed into those in Fig. 7, which suggests the presence of a minimum critical value of the locally de ned Reynolds number R, although it may be located slightly outside the convergence limit  ; 2 of the series solution. It is interesting and even instructive to compare the last result with Fig. 11 of the previous study (Itoh, 1996b) obtained from a simple model of disturbance equations, because it suggests a usefulness of modeled stability equations. Finally, we investigate critical Reynolds numbers of general Falkner–Skan–Cooke boundary layers with the parameter m 6= 1 at the limit of R1 in nity, for simplicity, by solving the homogeneous and inhomogeneous equations (4.4). The leading terms of the series solution (4.2) are obtained again from an eigenvalue problem of the lowest-order equations and the eigensolutions give variations of the critical Reynolds number and the corresponding frequency and wavenumbers with the Falkner– Skan pressure parameter m, as shown in Fig. 8. In order for the eigensolutions to be meaningful, however, the series solution (4.2) must converge for a given value of the Falkner–Skan parameter m. If the expansion parameter 1=R21 is replaced with  ≡ (1 − m)=2 in Eq. (5.1) and the resulting

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Fig. 6. Variations of the critical Reynolds number R0 of the attachment-line ow with the chordwise distance R1 for dierent values of the truncation number N .

Fig. 7. Critical values of the local Reynolds number R plotted against the inclination angle  for dierent values of the truncation number N .

equation is used in the same manner as before, we obtain the approximate critical Reynolds number R0 as a function of the parameter m and the truncation number N . Computational results under four dierent levels of approximation are presented in Fig. 9, where the four curves start from the critical point of the attachment-line ow at m = 1 and the curve of N = 0 denoting results of the lowest-order eigenvalue problem slightly deviates from the others as m decreases. However, the dierence of the three curves for N = 1; 2 and 3 from each other is very small if m is larger than about 0.6 and seems to indicate a very good convergence of the series solution in the same parameter region. These computational results show an important property of streamline-curvature instability that the critical Reynolds number near the attachment line decreases almost linearly with

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Fig. 8. The critical Reynolds number, frequency and wavenumbers of Falkner–Skan–Cooke ows obtained from the lowest-order eigenvalue problem.

Fig. 9. Variations of the critical Reynolds number R0 of Falkner–Skan–Cooke ows with the pressure parameter m for dierent values of N and for R1 in nity.

1 − m, but have failed to con rm a natural deduction that the critical value goes up as m approaches zero, because the streamline-curvature instability will disappear for weak pressure gradient (see Itoh, 1994b). It may be emphasized here that the rather wide region of convergence suggested will aord a mathematical basis for the rst approximation by solutions of the lowest-order eigenvalue problem to stability characteristics of a similar boundary-layer ow with the parameter m slightly less than unity, although further investigations are necessary for full assurance of the convergence.

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6. Concluding remarks A non-parallel theory has been developed to describe linear stability of three-dimensional Falkner– Skan–Cooke boundary layers to oblique-wave disturbances. In such similar boundary-layer ows, the dimensionless coordinate in the chordwise direction perpendicular to the attachment line is coincident with the local Reynolds number de ned by the chordwise velocity UE of external stream and the boundary-layer thickness  as R1 =UE =, and then the linearized disturbance equations are written in a partial dierential form with respect to the Reynolds number R1 and the dimensionless coordinate  in the normal-to-wall direction. These equations enable us to seek a solution of double power series in the two parameters 1=R21 and  ≡ (1 − m)=2; m denoting the Falkner–Skan pressure parameter, and in particular for the attachment-line ow with m = 1, the expansion parameter  vanishes and the solution is reduced to a single power series of 1=R21 , whose coecients are determined by solving a sequence of ordinary dierential systems. Numerical estimation of the rst few coecients has been made to clarify several fundamental properties of the series solutions. In the most important case of attachment-line ow with m=1, the series solution seems to converge for smaller values of R0 =R1 than about 0.5 and then eigensolutions of the lowest-order equation system are guaranteed to be the rst approximation to the true solution. Numerical estimation of the rst few coecients of the series implies that the critical Reynolds number of this ow to oblique-wave disturbances slowly varies from R0 = 87:1 at R1 = ∞ to R0 = 125 at R1 = 250 but the divergence of the series solution for lower values of R1 prevents us from revealing how this critical curve relates to the critical Reynolds number R0 = 583 of the spanwise Tollmien–Schlichting instability. It is also found that the oblique-wave disturbances appearing in this ow consist of two dierent modes; one is induced by the streamline-curvature instability and governs the low Reynolds number region with R0 less than 110, while the other results from the familiar cross- ow instability and becomes dominant in the region of higher Reynolds numbers. For larger values of R0 than 583, the Tollmien–Schliching waves propagating in the spanwise direction are added to the two oblique modes, so that the ow eld of such a high Reynolds number may be very complicated by their superposition even in the linear stage of disturbance development. A key to solve such a multi-instability problem may be found in the numerical method of stability calculation recently developed by Lin and Malik (1996). To investigate stability of swept Hiemenz ow to oblique Tollmien–Schlichting waves at suciently high Reynolds numbers, they expanded an eigensolution of the partial-dierential disturbance equations into Chebyshev polynomials of the normal coordinate  and power series of the chordwise coordinate R1 . The power series of R1 is valid for a region near the attachment line R1 = 0 but not for very large values of R1 in contrast to the present formulation based on the expansion about R1 = ∞, and the dierent regions of validity will shed light on dierent aspects of the true solution of the same disturbance equations. Although their computational results gave no information of instability phenomena other than Tollmien–Schlichting mode, we may expect that a careful modi cation of the expansion in power series of R1 with particular attention to multiple-instability characteristics of the

ow will provide us with some information about the cross- ow and streamline-curvature instability modes near the attachment line R1 = 0 that has not been obtained in the present study. The attachment-line boundary layer is the most important ow in the sense that it leads to the simplest form of non-parallel stability theory for growing boundary-layer ows. If we consider a more practical ow on an in nite swept wing, velocity distribution of the basic laminar ow deviates from that of the attachment-line ow with m = 1 as the chordwise distance R1 increases and a modi cation

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of the non-parallel theory is necessary for its application to general Falkner–Skan–Cooke boundary layers with m 6= 1. A double power-series solution can be derived on the basis of a false-expansion method, where a small parameter  denoting the deviation of the basic ow from the attachment-line

ow is taken as the second expansion parameter added to the rst one 1=R21 . Results of numerical computations for the limiting case of R1 in nity indicate that the solution expanded into powers of  seems to converge if  is less than about 0.2, so that eigensolutions of the leading equation system may be considered to present the rst approximation to the principal stability characteristics of the three-dimensional similar boundary layers with larger values than about 0.6 of the Falkner–Skan pressure parameter m. It is worthwhile to note that the present theory includes the most important non-parallel terms associated with the vertical component of the basic boundary-layer ow in the stability equations at the lowest order (see also Itoh, 1998). On most swept wings of conventional high-speed aircraft, the boundary-layer instability and transition occurs in the wind-facing region near the leading edge. Experimental observations by Poll (1979) identi ed the spanwise-traveling Tollmien–Schlichting waves in the attachment-line boundary layer on a yawed circular cylinder, while Takagi and Itoh (1998), Tokugawa et al. (1999) experimentally investigated the development of both cross- ow and streamline-curvature modes in a fully developed boundary layer slightly downstream on a yawed cylinder. For physical understanding of the early transition in swept-wing boundary layers, however, more detailed experiments should be conducted on stability characteristics of the ow eld between the attachment line and the downstream region. It is expected that results of the present study will contribute to a success in such careful and minute observations to be made in the near future. Acknowledgements The author wishes to express his thanks to Drs. S. Takagi and N. Tokugawa for continuous and helpful discussion on the subject during the course of this study. References Balakumar, P., Trivedi, P.A., 1998. Finite amplitude stability of attachment line boundary layers. Phys. Fluids 10, 2228–2237. Hall, P., Malik, M.R., Poll, D.I.A., 1984. On the stability of an in nite swept attachment line boundary layer. Proc. Roy. Soc. Lond. A 395, 229–245. Itoh, N., 1994a. Centrifugal instability of three-dimensional boundary layers along concave walls. Trans. Japan Soc. Aero. Space Sci. 37, 125–138. Itoh, N., 1994b. Instability of three-dimensional boundary layers due to streamline curvature. Fluid Dyn. Res. 14, 353–366. Itoh, N., 1996a. Streamline-curvature instability of three-dimensional boundary layers: Part II. Stability estimation of the

ow on a yawed elliptic cylinder. Trans. Japan Soc. Aero. Space Sci. 39, 60 –73. Itoh, N., 1996b. Simple cases of the streamline-curvature instability in three-dimensional boundary layers. J. Fluid Mech. 317, 129–154. Itoh, N., 1998. Eects of non-parallelism on Gortler instability. Proc. 30th Symposium on Turbulence, ed. Japan Soc. Fluid Mech., pp. 161–162 (in Japanese). Lin, R.S., Malik, M.R., 1996. On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz ow. J. Fluid Mech. 311, 239–255. Poll, D.I.A., 1979. Transition in the in nite swept attachment-line boundary layer. Aeronaut. Quart. 30, 607–629.

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Radeztsky, R.H., Reibert, M.S., Saric, W.S., 1994. Development of stationary cross ow vortices on a swept wing. AIAA Pap. No. 94-2373. Reed, H.L., Saric, W.S., 1989. Stability of three-dimensional boundary layers. Ann. Rev. Fluid Mech. 21, 235–284. Reed, H.L., Saric, W.S., Arnal, D., 1996. Linear stability theory applied to boundary layers. Ann. Rev. Fluid Mech. 28, 389 –428. Rosenhead, L., 1963. Laminar Boundary Layers. Oxford Univ. Press, Oxford (Chapter 8). Takagi, S., Itoh, N., 1998. Dispersive evolution of cross ow disturbances excited by an airjet column in a three-dimensional boundary layer. Fluid Dyn. Res. 22, 25– 42. Tokugawa, N., Takagi, S., Itoh, N., 1999. Excitation of streamline-curvature instability in three-dimensional boundary layer on a yawed cylinder. AIAA Pap. No. 99-0814.