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On Nonlinear Evolution of C-type Instability in Nonparallel Boundary Layers
Chinese
Chinese Journal of Aeronautics 20(2007) 313-319
Journal of Aeronautics
www.elsevier.com/locate/cja
On Nonlinear Evolution of C-type Instability in Nonparallel Boundary Layers Liu Jixuea,b, Tang Dengbin a,*, Yang Yingzhaoa a
Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b
Chengdu Aircraft Design Institute, Chengdu 610041, China Received 30 October 2006; accepted 16 April 2006
Abstract The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations (PSE). Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results. Keywords: C-type instability; nonlinear evolution; nonparallelism; boundary layer stability; parabolic stability equations
1 Introduction* The boundary layer transition problem from the laminar to the turbulent flows is closely related with the vehicle design in the aviation and aerospace. Then nonlinear evolution of instability flow in boundary layers is the most important stage of the transition process, and extremely complicated. Klebanoff and others[1] found in the stability research of boundary layers that although flow instability begins with two-dimensional (2-D) disturbance, three-dimensional (3-D) disturbance appear soon, and their amplitudes grow quickly. As a result, the flow undergoes a change from the laminar to the turbulent. ‘K-type’ instability together with the C-type[2] and H-type[3] instability constitute tree*
Corresponding author. Tel.: +86-25-88489767. E-mail address: [email protected] Foundation item: Doctoral Foundation of Ministry of Education of China (20030287003)
type route to the transition. In the C-type instability, 3-D mode with half frequency of the TS wave (subharmonic), and the peak-valley splitting structure are different from that with the frequency equal to the TS wave and the peak-valley arranged structure in the K-type instability. Which one of types of instability will appear bears direct relation to the magnitude of initial amplitude of disturbances. In the study of instability in nonparallel boundary layers[4], the nonlinear evolution of the boundary layer flow needs to be taken into simultaneous consideration. This is a very complex and difficult problem. Until now, the useful method is limited. The newly developed method called the parabolic stability equations[5] seems to be helpful to tackle this problem. As a matter of fact, streamwise changes of the velocity profiles, wavelengths and the growth rates of disturbances are so slow that their second derivatives and products of first deriva-
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tives are sufficiently small to be neglected. Thus the parabolic stability equations can be derived from the N-S equations. It is noticeable that the derivation of PSE is not confined by the amplitude of the disturbance, so nonparallel and nonlinear terms can be incorporated in the marching procedure used to solve the parabolic stability equations[6]. Therefore, this method is especially suitable for studying nonlinear problem of ‘C-type’ instability in nonparallel boundary layers.
2 Nonlinear Parabolic Stability Equations The disturbances are introduced into the N-S equations, and, by eliminating the velocity component in the z-direction, the governing equations of the disturbances in the form of u, v, which are in xand y-direction, respectively, are obtained. For a given TS wave of frequency 2F, streamwise wavenumber 2α, spanwise wavenumber β and phase speed c, the disturbance flow field can be expanded into Fourier components having frequencies nF (n = 0, 1, 2, ···) and spanwise wavenumbers kβ (k = 0, 1, 2, ···). Then for a wave with the frequency not equal to zero, these components possess a phase speed equal to that of TS wave. The velocities of flow field can be represented as following form: u=u +
∞
∞
⎫
∑ ∑ u( n,k ) ( x, y )⎪
⎪ ⎪ v = v + ∑ ∑ v( n,k ) ( x, y ) ⎪⎬ (1) n =−∞ k =−∞ ⎪ ⎪ u( n, k ) = uˆ( n, k ) χ( n, k ) ⎪ v( n, k ) = vˆ( n, k ) χ ( n,k ) ⎪⎭ in which uˆ( n, k ) , vˆ( n, k ) are the shape function of the n =−∞ k =−∞ ∞
∞
mode (n, k) in x- and y-direction, respectively, and χ( n, k ) can be expressed as
χ( n, k ) = exp
(∫ α x
x0
( n, k )
(ξ ) dξ + ik β z − iωt
)
(2)
where a(n,k ) (x) = γ (n,k ) (x) + inα(x) , α ( x) is the streamwise wave number, and γ ( n, k ) ( x) is the exponential growth rate of disturbances. According to parabolic stability theory, assuming that the variations of velocity profile, wavenumber and growth rate are very slow in the streamwise
streamwise direction. Hence, the streamwise derivatives of u( n, k ) , v( n, k ) of the mode (n, k) of three-dimensional disturbances in the boundary layers take the following simplified form: ∂ m u( n, k ) ⎡ m ∂uˆ( n,k ) ⎫ m −1 ⎪ ˆ = + a u ma ⎢ ( n, k ) ( n, k ) ( n, k ) ∂x + ⎪ ∂x m ⎢⎣ ⎪ da( n, k ) ⎤ ⎪ m m−2 uˆ( n,k ) ⎥ χ( n, k ) ⎪ ( m − 1) a( n,k ) 2 dx ⎪ ⎦⎥ (3) ⎬ m ∂ v( n,k ) ⎡ m ∂vˆ( n, k ) ⎪ m −1 = ⎢ a( n,k ) vˆ( n, k ) + ma( n, k ) + ⎪ ∂x ∂x m ⎪ ⎣⎢ ⎪ da n, k ⎤ m ( m − 1) a(mn−,k2) ( ) vˆ( n,k ) ⎥ χ( n,k ) ⎪⎪ 2 dx ⎥⎦ ⎭ After introducing the above expressions into the disturbance equations, and collecting the terms multiplied by exponential functions of the same frequency nω and spanwise wavenumber kβ, and let da/dx =0, a set of coupled nonlinear partial differential equations of 3-D disturbances are obtained as ⎡ ∂uˆ( n, k ) ⎤ L(1)u ⎡uˆ( n, k ) ⎤ + L(1) v ⎡ vˆ( n, k ) ⎤ + M (1)u ⎢ ⎥+ ⎣ ⎦ ⎣ ⎦ ⎣⎢ ∂x ⎦⎥
⎫ ⎪ ⎪ ⎪ ∞ ∞ ⎡ ∂vˆ( n, k ) ⎤ ⎪ (1) (1) M v⎢ ⎥ = ∑ ∑ N ( i, j , n − i, k − j ) ⎪ x ∂ ⎪ ⎣⎢ ⎦⎥ i =−∞ j =−∞ ⎬ (4) ⎡ ∂uˆ( n, k ) ⎤ ⎪ (2) ⎡ (2) (2) L u uˆ( n, k ) ⎤ + L v ⎡ vˆ( n, k ) ⎤ + M u ⎢ ⎥+ ⎪ ⎣ ⎦ ⎣ ⎦ x ∂ ⎣⎢ ⎦⎥ ⎪ ⎪ ∞ ∞ ⎡ ∂vˆ( n, k ) ⎤ (2) ⎪ M (2) v ⎢ ⎥ = ∑ ∑ N ( i, j , n − i, k − j ) ⎪ x ∂ ⎣⎢ ⎦⎥ i =−∞ j =−∞ ⎭
where L(1)u = −2a( n, k )
∂u ∂ ∂ 2u − 2a( n, k ) ∂x ∂y ∂x∂y
1 ∂4 2 2 ∂3 ⎡ + v 3 + ⎢ a( n, k ) u − inω − a( n, k ) − 4 R0 ∂y R ∂y ⎣ 0 ∂u ⎤ ∂ 2 ⎡ ∂ 2u ⎤ ∂ k 2 β 2 ) ⎥ 2 + ⎢v a(2n, k ) − k 2 β 2 − ⎥ + ∂x ⎦ ∂y ∂x∂y ⎦ ∂y ⎣
(
L(1) v = −
(
( a(
− k2β 2
( a(
⎡ ⎤ 1 2 − k 2 β 2 ⎢ a( n, k ) u − inω − a( n, k ) − k 2 β 2 ⎥ R0 ⎣ ⎦
2 n, k ) 2 n, k )
) ∂∂ux − a(
)
n, k )
∂ 2u ∂y 2
)
−
∂ 3u ∂x∂y 2
+
(
)
M (1)u = 0
M (1) v = u
∂2 ∂y
2
(
)
− 2ia( n,k ) nω + u 3a(2n, k ) − k 2 β 2 −
∂ 2u ∂y 2
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
(
( a( ) ) − k β 2 n,k
2
L(2) v =−
)
2
(
α ( n,k ) ∂ 3 R0 ∂y
3
+ α ( n,k ) v
⎡ ⎣
M
u
)
⎡ 1 2 ∂v ⎤ a( n, k ) − k 2 β 2 − ⎥ ⎢ a( n,k ) u − inω − R0 ∂y ⎦ ⎣
α ( n, k ) ⎢α ( n, k ) u − inω − (2)
(
1 2 ∂2 ∂ a( n,k ) − k 2 β 2 + v a(2n, k ) − k 2 β 2 + 2 R0 y ∂ ∂y
L(2)u = −
(
∂2 ∂y
2
− k 2β 2
)
∂u + ∂y
⎤ ∂ ∂v 1 (α (2n, k ) − k 2 β 2 ) ⎥ − ∂y R0 ⎦ ∂y
)
= u 3a(2n, k ) − k 2 β 2 − 2ia( n, k ) nω
(
)
∂ M (2) v = 2a( n, k ) u − inω ∂y
The nonlinear terms in the right side of Eq.(4) constitute an extremely complicated nonlinear function in disturbance waves[7]. The boundary conditions are ∂ ⎫ u( n, k ) ( x, 0 ) = 0, v( n,k ) ( x, 0 ) = v( n,k ) ( x, 0 ) = 0 ⎪ ∂y ⎪ ⎬ (5) ∂ u( n, k ) ( x, ∞ ) = 0, v( n,k ) ( x, ∞ ) = v( n,k ) ( x, ∞ ) = 0 ⎪ ∂y ⎭⎪ Since governing Eq.(4) is partial differential system with an unknown parameter a(x), another condition is needed to make the solution unique. In order to make uˆ( n, k ) ( x, y ) , vˆ(n ,k ) (x, y ) and α (n, k ) (x ) satisfy the hypothesis of PSE, their variations are limited in the x-direction, alluding to normalization condition. Here selected the form as follows: ∂uˆ( n, k ) ∞ + ⎫ ∫0 uˆ(n,k ) ∂x dy = 0, for n = 2, k = 0 ⎪⎪ (6) ⎬ ⎪⎧ ∞ + ∂uˆ( n, k ) ⎪⎫ ⎪ Re ⎨ ∫ uˆ( n, k ) dy ⎬ = 0, for other modes 0 ⎪ ∂x ⎩⎪ ⎭⎪ ⎭ where, “+” denotes complex conjugate. The normalization for n = 0, k = 2 supplies two scalar equations to the determination of γ(2,0) and α. The other normalization conditions can supply one scalar equation for the unknown growth rate γ(n,k). Thus Eqs.(4)-(6) constitute nonlinear parabolic stability equations.
3 Initial Conditions and Analysis of Modes 3.1 Initial conditions The initial conditions are composed by the TS wave, mean-flow-distortion, subharmonic wave and
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the harmonics of TS wave. While the TS wave and the subharmonic wave can be defined by the results of the linear PSE. The complicated mean-flow-distortion and harmonics will be studied as follows spectively. The mean-flow-distortion (0, 0) is determined by the quadratic interaction between the TS wave and itself, and between other harmonics, thus the mean-flow-distortion will have a spatial growth rate approximately twice that of the TS wave. By denoting this exponential growth rate by α0, the calculation of the mean-flow-distortion is calculated approximately with the help of the spatial growth concept in the nonparallel basic-flow. In order to obtain an ordinary differential equation, the profiles of the mean-flow-distortion, the TS wave and the harmonics are approximatly supposed to be locally selfsimilar, e.g.,
ψ n = δ ( x) f n (η ) exp[nα n x − inωt ] n = 0,1, 2, ⋅⋅⋅
(7)
Furthermore, we can get u=
∂ψ B n =∞ n + ∑ A ⋅ f n′ ⋅ exp[α n x − inω t ] ⎫ ⎪ ∂y n =−∞
⎪ ⎪⎪ ∂ψ B n =∞ n δ ( x) y − ∑ A ⋅[ v=− fn − f n′ + ⎬ ⎪ ∂x n =−∞ 2x 2x ⎪ n≠0 ⎪ δ ( x) ⋅ f n ⋅ α n ] ⋅ exp[α n x − inω t ] ⎪⎭ n≠0
(8)
where η = y / δ ( x) , f n (η ) = φn ( x0 , y ) , and the exponent contains α n , which is the growth rate and wavenumber based on u'max. By substituting Eq.(7) into the u-component of the N-S equations, and coupling Eq.(8) and omitting the terms of order o(R–2), the following equation is yielded −
a02 1 f 0′ + a0 ( f 0′F ′ − f 0 F ′′) − (2 f 0′′′+ Ff 0′′ + R 2R F ′′f 0 ) =
∞
∑ an fn f−′′n − a− n f −′n f n′ n =−∞
(9)
n≠0
in which the terms of order R–1 on the right-hand side have been omitted, since the approximation of self-similar flow in Eq.(7) produces errors of the same order. The basic-flow streamfunction is denoted by F(η), where F is obtained by solving the Blasius equation.
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The higher-order harmonics are created by nonlinear interaction between lower-order harmonics, and it is supposed at beginning to be α n = nα1 , n > 1 (10) where, the unknowns at this point are the profiles φn and the exponent α1. Then each of these quantities (such as α1, ϕ1, ϕ2, etc.) is expanded in a Taylor series in the amplitude A. The quadratic nonlinearity of the N-S equations leads to expansions in even exponents of A, and the flow field becomes ψ ( x, y, t , A) = ψ B ( x, y ) + A2 (φ01 + A2φ02 + A4φ03 + ⋅⋅⋅) ⋅ exp(α 0 x) + A(φ10 + A2φ11 + A4φ12 + ⋅⋅⋅) exp[(α10 + A2α11 + A4α12 + ⋅⋅⋅) x − iωt ] + A2 (φ20 + A2φ21 + A4φ22 + ⋅⋅⋅) exp[2(α10 + A2α11 + A4α12 + ⋅⋅⋅) x − i2ωt ] + A3 (φ30 + A2φ31 + A4φ32 + ⋅⋅⋅) exp[3(α10 + A2α11 + A4α12 +
) x − i3ωt ]
(11)
By collecting terms multiplied by equal powers of A and arranging the governing equations in hierarchical order, and unknowns may be found out. Consistent with the definition of A based on the value of the maximum u velocity of the TS wave, the normalizations condition is chosen as follows, ∂φ ∂φ ( y ) ⎫ max( 10 ) = 10 m ⎪ ∂y ∂y ⎪ (12) ⎬ ∂φ1k ( ym ) ⎪ = 0, k = 2,3, 4, ⋅⋅⋅ ⎪⎭ ∂y in which ym is the position where the maximum of the u profile of the TS wave occurs. At orders A1, A2, ···, φ1n, α1n are obtained by solving simultaneously local partial differential equations with the normalization condition. 3.2 Analysis of modes A series of new modes will be generated in the process of C-type transition by the nonlinear interaction between the TS wave and the subharmonic wave, and the harmonic wave of TS wave. Next we analyze these new modes. The initial conditions are taken as the fundamental of the TS wave denoted by ‘T’, and the subharmonic denoted by ‘C’. With these waves being assumed to have small finite amplitude, the results of analysis are listed in Table 1. As the amplitude increases, the interaction between harmonics waves becomes so large as to gen-
erate new set of modes shown in Table 2. The small letters in the table indicate the nonlinear product giving rise to the modes. Table 1 Analysis of modes at initial condition n=0
1
2
k=0
T
k=1
C
Table 2 Analysis of new modes generated by nonlinear effects n=0 k=0 k=1 k=2
1
t2
2
c c2
3
4 c2
T ct c2
c2t
In this paper the matrix has been truncated to n = 4 and k = 2, i.e., n ∈ [−4, 4] , k ∈ [−2, 2] . The disturbance modes and the initial order given in Table 2, a set of (n, k) in the flow field has been analyzed.
4 Numerical Method 4.1 High-order spectral method and mapping In order to improve the calculation precision during solution of PSE, the high-order spectral method is used. With the Chebyshev functions Tn(y) of the first kind of Chebyshev polynomials chosen, forty Chebyshev polynomials are adopted in this paper. The approximate solution is written as following 40
40
0
0
u( n, k ) = ∑ a( n, k )mTm ( y ), v( n, k ) = ∑ b( n, k )mTm ( y ) (13)
In the calculation, the mapping is needed from the semi-infinite physical domain y ∈ [0, ∞) to calculated domain Y ∈ [−1, +1] . Here is employed algebraic mapping, because of its accuracy higher than exponential mapping[8]. It may be written in the form of Y = (by − y0 ) /(by + y0 ) . The differential operators are converted into algebraic form by aid of a spectral collocation method. Where b=1, and y0 controlling the collocation in the physical domain. Here is obtained an algebraic system, which might be solved with the help of the QR matrix algorithm. 4.2 Procedure of solving the PSE The complex velocity profiles satisfy the relationships, uˆ(n ,k ) = uˆ(n , − k ) , uˆ( n,k ) = uˆ(+− n, k ) , which are
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
based on physical characteristics of the flow-field. And, for vˆ(n ,k ) , wˆ (n,k ) , the relations can be written the same way. Thus only the modes having index n≥0, k≥0 need to be solved and stored in memory, causing a decreased number of unknowns. In using the method above-mentioned, initial conditions are given: the TS wave (2, 0), the harmonics (4, 0), the mean-flow-distortion (0, 0) and the subharmonic wave (1, 1). In the marching process of solving nonlinear PSE, firstly the solution of TS wave at each station is to be found by iterating and updating growth rate γ(2,0) and wavenumber αˆ (2,0) , and satisfying the normalization condition. Here comes the following iteration formula
(
⎡ ∞ i i Δγ ((n), k , m +1) = Re ⎢ ∫ uˆ((n),k ,m +1) + uˆ( n, k , m ) ⎣ 0
(
+
)
i uˆ((n), k , m +1) − uˆ( n, k , m ) / ( 2Δx ) dy ⎤ ⎥⎦
(
⎡ ∞ i) i) Δαˆ((2,0, = Im ⎢ ∫ uˆ((2,0, + uˆ( 2,0, m ) m +1) m +1) ⎣ 0
(
)i
)
i) − uˆ( 2,0, m ) / ( 2Δx ) dy ⎤ uˆ((2,0, m +1) ⎥⎦
)i
(14)
+
(15)
where the superscript ‘i’ denote i-th iteration, the subscript ‘m’ indicates the streamwise step number and xm = x0 + m Δ x . The ‘Re’ and ‘Im’ denote the real and the image of the parameter, respectively. The γ is growth rate, and α is wavenumber of the TS wave at this station. The number of iteration used at each step varies with the rate of convergence of γ and α. After streamwise wavenumber of the TS wave is obtained, that of other harmonics can be determined nαˆ( 2,0, m +1) αˆ( n, k , m +1) = (16) 2 Thus, the all harmonics at (m+1) station have been calculated, it is possible to march to next station by using the same method.
5 Nonlinear Evolution Analysis of C-type Instability Now the three-dimensional nonlinear C-type instability in the Blasius boundary layer is to be studied. In order to compare the solutions of the
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nonlinear PSE to the experimental data given by Saric[9], let F=106, b=0.2 in marching process of calculation, in which non-dimensional frequency F= 106ω/Re0, the non-dimensional wavenumber in the spanwise direction b=103β/R0. The velocity is nondimensionalized by using UB, and the length by δ 0 = (ν x0 / U 0 )1/ 2 , where x0 is a fixed dimensional distance from the leading edge, ν is the kinematical viscosity and R0=UBδ0/ν. The computational results of the PSE and the experimental data are graphed in the Fig.1, where the evolution curves of the amplitudes of two-dimensional wave jumps up suddenly in the vicinity of Re=688 on the experimental curves, which is ascribed to the interference of the smoke device[9], and then revert to normally at Re=700. Under this circumstance, the flow region is divided into two sections in the computation, i.e. section 1 (Re= 590-685) and section 2 (Re=700-750). The initial amplitude of the TS wave is A0=0.001 8 in section 1, and A0=0.006 4 in section 2. The initial amplitude of the subharmonics (C-Type) is B0=0.001 4. The initial values of other harmonics, which are generated by the nonlinear interactions of these two type waves, are also decided by these interactions. It is thus clear from the figure that amplitude curve of the TS wave (2, 0) has good agreement with experiment data in section 1 and section 2 after the nonlinear effects have been taken into consideration during calculation. In the process of marching and enlarging gradually along stream direction, nonlinear influences of the disturbances are also obvious. Consequently, without consideration of these influences, the results can only accord the experiment date at the position having relatively smaller Reynolds number such as in section 1, while at the position with bigger Reynolds number like in section 2 significant errors will appear[10]. The evolution curve of the amplitudes for 3-D subharmonics (1, 1) is also given in this figure. The variety of the curve is very slow at Re < 690, but at Re > 700, the variety is quick. Then growth rate is greater than that of the TS wave, and nonlinear affection is dominated, and results computed are consistent with
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Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
experiment date. The evolution curves of modes (2, 2), (3, 1) and (4, 2) produced by nonlinear interaction are shown in Fig.2. From the figure, it follows that the nonlinear influences of these modes increases with the rise of Reynolds number, especially when Re > 700. It is the significant characteristic feature that must be paid a close attention.
Fig.1
Fig.2
Fig.3
Profile curve of mode (0,0).
Fig.4
Profile curve of mode (2,2).
Fig.5
Profile curve of mode (3,1).
Fig.6
Profile curve of mode (4,0).
Amplitude curves of modes (2,0) and (1, 1) in C-type instability.
Amplitude curve of modes (2,2), (3,1) and (4,2) in C-type instability.
The profiles of the velocity and the evolution problems related to different modes are analyzed below. In order to discuss for the convenience, the results in Figs.3-6 are obtained at the same Reynolds number (Re=638). It is obvious from the figures that the maximum value and the corresponding position for each modes generated by the nonlinear interaction of disturbance waves are obviously different, and the shape and the trend of the variation also have the remarkable dissimilarity. Fig.7 presents the profiles of mode (2, 0) at two different Reynolds numbers. By comparing the results at Re-
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
ynolds number (Re=736) to those at the Reynolds number (Re=638) in Fig.7, we find that the maximum value of the profile at Re=736 is bigger, and its position is farther away from the wall. Furthermore, the three-dimensional evolution of profiles in stream direction at different Reynolds number for the subharmonic (1, 1) is shown in Fig.8. Obviously, the maximum values of profiles are increased continuously, and this growth is not quick at the linear stage at the smaller Reynolds number, but when developing further downstream, the nonlinear function increases evidently, and the maximum value enlarge rapidly, and its position is farther gradually away from the wall. Obviously, the nonlinear evolution process is influenced mainly by the Reynolds number.
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harmonic wave in the C-type instability of the threedimensional disturbance is studied very effectively by PSE method and the numerical techniques developed in the paper. The different results of various modes generated by the nonlinear interaction are obtained, and Reynolds numbers are pointed out to be the decisive factor. It is proved that the presented method is in strict conformity with the experimental data. References [1]
Klebanoff P S, Tidstorm K D, Sargent L M. The three-dimensional nature of boundary layer instability. J F M 1962; 12: 1-34.
[2]
Craik A A D. Non-linear resonant instability in boundary layers. J F M 1971; 50: 393-415.
[3]
Herbert T. Secondary instability plane channel flow to subharmonic three dimensional disturbance. Physics of Fluids 1983; 26: 871-874.
[4]
Tang D B, Ma Q R, Cheng G W. Study of compressible nonparallel boundary layers. Acta Aeronautica et Astronautica Sinica 2002; 23(2): 166-169. [in Chinese]
[5]
Herbert T. On the stability of 3D boundary layers. AIAA 97-1961, 1997.
[6]
Tang D B, Xia H. Nonlinear evolution analysis of T-S disturbance wave at finite amplitude in nonparallel boundary layers. Applied Mathematics and Mechanics 2002; 23(6): 660-669.
[7]
Wang W Z, Tang D B. Studies on nonlinear stability of three-dimensional H-type disturbance. Acta Mechanica Sinica
Fig.7
2003; 19(6): 517-526.
Profile curves of TS wave. [8]
Grosch C E, Orszag S A. Numerical solution of problem in unbounded regions: coordinate transforms. Journal of Computational Physics 1997; 25: 273-296.
[9]
Saric W S, Kozlor V V, Levehenko V Y. Forced and unforced subharmonic resonance in boundary layer transition. AIAA-840007, 1984.
[10]
Lu C G, Zhao G F. Theoretical analysis of C-type instability in the Blassius boundary layer. Aerodynamica Sinica 1995; 13(3): 329-333. [in Chinese]
Biography: Liu Jixue Fig.8
Profile curves of mode (1,1) at different Reynolds numbers change with stream direction.
6 Conclusions Nonlinear evolution process of TS wave and
Born in 1979, he received B.S.
and M.S. from Nanjing Aeronautical Institute in 2003 and 2006 respectively, and then became an engineer in Chengdu aircraft design institute.
Chinese Journal of Aeronautics 20(2007) 313-319
Journal of Aeronautics
www.elsevier.com/locate/cja
On Nonlinear Evolution of C-type Instability in Nonparallel Boundary Layers Liu Jixuea,b, Tang Dengbin a,*, Yang Yingzhaoa a
Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b
Chengdu Aircraft Design Institute, Chengdu 610041, China Received 30 October 2006; accepted 16 April 2006
Abstract The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations (PSE). Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results. Keywords: C-type instability; nonlinear evolution; nonparallelism; boundary layer stability; parabolic stability equations
1 Introduction* The boundary layer transition problem from the laminar to the turbulent flows is closely related with the vehicle design in the aviation and aerospace. Then nonlinear evolution of instability flow in boundary layers is the most important stage of the transition process, and extremely complicated. Klebanoff and others[1] found in the stability research of boundary layers that although flow instability begins with two-dimensional (2-D) disturbance, three-dimensional (3-D) disturbance appear soon, and their amplitudes grow quickly. As a result, the flow undergoes a change from the laminar to the turbulent. ‘K-type’ instability together with the C-type[2] and H-type[3] instability constitute tree*
Corresponding author. Tel.: +86-25-88489767. E-mail address: [email protected] Foundation item: Doctoral Foundation of Ministry of Education of China (20030287003)
type route to the transition. In the C-type instability, 3-D mode with half frequency of the TS wave (subharmonic), and the peak-valley splitting structure are different from that with the frequency equal to the TS wave and the peak-valley arranged structure in the K-type instability. Which one of types of instability will appear bears direct relation to the magnitude of initial amplitude of disturbances. In the study of instability in nonparallel boundary layers[4], the nonlinear evolution of the boundary layer flow needs to be taken into simultaneous consideration. This is a very complex and difficult problem. Until now, the useful method is limited. The newly developed method called the parabolic stability equations[5] seems to be helpful to tackle this problem. As a matter of fact, streamwise changes of the velocity profiles, wavelengths and the growth rates of disturbances are so slow that their second derivatives and products of first deriva-
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tives are sufficiently small to be neglected. Thus the parabolic stability equations can be derived from the N-S equations. It is noticeable that the derivation of PSE is not confined by the amplitude of the disturbance, so nonparallel and nonlinear terms can be incorporated in the marching procedure used to solve the parabolic stability equations[6]. Therefore, this method is especially suitable for studying nonlinear problem of ‘C-type’ instability in nonparallel boundary layers.
2 Nonlinear Parabolic Stability Equations The disturbances are introduced into the N-S equations, and, by eliminating the velocity component in the z-direction, the governing equations of the disturbances in the form of u, v, which are in xand y-direction, respectively, are obtained. For a given TS wave of frequency 2F, streamwise wavenumber 2α, spanwise wavenumber β and phase speed c, the disturbance flow field can be expanded into Fourier components having frequencies nF (n = 0, 1, 2, ···) and spanwise wavenumbers kβ (k = 0, 1, 2, ···). Then for a wave with the frequency not equal to zero, these components possess a phase speed equal to that of TS wave. The velocities of flow field can be represented as following form: u=u +
∞
∞
⎫
∑ ∑ u( n,k ) ( x, y )⎪
⎪ ⎪ v = v + ∑ ∑ v( n,k ) ( x, y ) ⎪⎬ (1) n =−∞ k =−∞ ⎪ ⎪ u( n, k ) = uˆ( n, k ) χ( n, k ) ⎪ v( n, k ) = vˆ( n, k ) χ ( n,k ) ⎪⎭ in which uˆ( n, k ) , vˆ( n, k ) are the shape function of the n =−∞ k =−∞ ∞
∞
mode (n, k) in x- and y-direction, respectively, and χ( n, k ) can be expressed as
χ( n, k ) = exp
(∫ α x
x0
( n, k )
(ξ ) dξ + ik β z − iωt
)
(2)
where a(n,k ) (x) = γ (n,k ) (x) + inα(x) , α ( x) is the streamwise wave number, and γ ( n, k ) ( x) is the exponential growth rate of disturbances. According to parabolic stability theory, assuming that the variations of velocity profile, wavenumber and growth rate are very slow in the streamwise
streamwise direction. Hence, the streamwise derivatives of u( n, k ) , v( n, k ) of the mode (n, k) of three-dimensional disturbances in the boundary layers take the following simplified form: ∂ m u( n, k ) ⎡ m ∂uˆ( n,k ) ⎫ m −1 ⎪ ˆ = + a u ma ⎢ ( n, k ) ( n, k ) ( n, k ) ∂x + ⎪ ∂x m ⎢⎣ ⎪ da( n, k ) ⎤ ⎪ m m−2 uˆ( n,k ) ⎥ χ( n, k ) ⎪ ( m − 1) a( n,k ) 2 dx ⎪ ⎦⎥ (3) ⎬ m ∂ v( n,k ) ⎡ m ∂vˆ( n, k ) ⎪ m −1 = ⎢ a( n,k ) vˆ( n, k ) + ma( n, k ) + ⎪ ∂x ∂x m ⎪ ⎣⎢ ⎪ da n, k ⎤ m ( m − 1) a(mn−,k2) ( ) vˆ( n,k ) ⎥ χ( n,k ) ⎪⎪ 2 dx ⎥⎦ ⎭ After introducing the above expressions into the disturbance equations, and collecting the terms multiplied by exponential functions of the same frequency nω and spanwise wavenumber kβ, and let da/dx =0, a set of coupled nonlinear partial differential equations of 3-D disturbances are obtained as ⎡ ∂uˆ( n, k ) ⎤ L(1)u ⎡uˆ( n, k ) ⎤ + L(1) v ⎡ vˆ( n, k ) ⎤ + M (1)u ⎢ ⎥+ ⎣ ⎦ ⎣ ⎦ ⎣⎢ ∂x ⎦⎥
⎫ ⎪ ⎪ ⎪ ∞ ∞ ⎡ ∂vˆ( n, k ) ⎤ ⎪ (1) (1) M v⎢ ⎥ = ∑ ∑ N ( i, j , n − i, k − j ) ⎪ x ∂ ⎪ ⎣⎢ ⎦⎥ i =−∞ j =−∞ ⎬ (4) ⎡ ∂uˆ( n, k ) ⎤ ⎪ (2) ⎡ (2) (2) L u uˆ( n, k ) ⎤ + L v ⎡ vˆ( n, k ) ⎤ + M u ⎢ ⎥+ ⎪ ⎣ ⎦ ⎣ ⎦ x ∂ ⎣⎢ ⎦⎥ ⎪ ⎪ ∞ ∞ ⎡ ∂vˆ( n, k ) ⎤ (2) ⎪ M (2) v ⎢ ⎥ = ∑ ∑ N ( i, j , n − i, k − j ) ⎪ x ∂ ⎣⎢ ⎦⎥ i =−∞ j =−∞ ⎭
where L(1)u = −2a( n, k )
∂u ∂ ∂ 2u − 2a( n, k ) ∂x ∂y ∂x∂y
1 ∂4 2 2 ∂3 ⎡ + v 3 + ⎢ a( n, k ) u − inω − a( n, k ) − 4 R0 ∂y R ∂y ⎣ 0 ∂u ⎤ ∂ 2 ⎡ ∂ 2u ⎤ ∂ k 2 β 2 ) ⎥ 2 + ⎢v a(2n, k ) − k 2 β 2 − ⎥ + ∂x ⎦ ∂y ∂x∂y ⎦ ∂y ⎣
(
L(1) v = −
(
( a(
− k2β 2
( a(
⎡ ⎤ 1 2 − k 2 β 2 ⎢ a( n, k ) u − inω − a( n, k ) − k 2 β 2 ⎥ R0 ⎣ ⎦
2 n, k ) 2 n, k )
) ∂∂ux − a(
)
n, k )
∂ 2u ∂y 2
)
−
∂ 3u ∂x∂y 2
+
(
)
M (1)u = 0
M (1) v = u
∂2 ∂y
2
(
)
− 2ia( n,k ) nω + u 3a(2n, k ) − k 2 β 2 −
∂ 2u ∂y 2
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
(
( a( ) ) − k β 2 n,k
2
L(2) v =−
)
2
(
α ( n,k ) ∂ 3 R0 ∂y
3
+ α ( n,k ) v
⎡ ⎣
M
u
)
⎡ 1 2 ∂v ⎤ a( n, k ) − k 2 β 2 − ⎥ ⎢ a( n,k ) u − inω − R0 ∂y ⎦ ⎣
α ( n, k ) ⎢α ( n, k ) u − inω − (2)
(
1 2 ∂2 ∂ a( n,k ) − k 2 β 2 + v a(2n, k ) − k 2 β 2 + 2 R0 y ∂ ∂y
L(2)u = −
(
∂2 ∂y
2
− k 2β 2
)
∂u + ∂y
⎤ ∂ ∂v 1 (α (2n, k ) − k 2 β 2 ) ⎥ − ∂y R0 ⎦ ∂y
)
= u 3a(2n, k ) − k 2 β 2 − 2ia( n, k ) nω
(
)
∂ M (2) v = 2a( n, k ) u − inω ∂y
The nonlinear terms in the right side of Eq.(4) constitute an extremely complicated nonlinear function in disturbance waves[7]. The boundary conditions are ∂ ⎫ u( n, k ) ( x, 0 ) = 0, v( n,k ) ( x, 0 ) = v( n,k ) ( x, 0 ) = 0 ⎪ ∂y ⎪ ⎬ (5) ∂ u( n, k ) ( x, ∞ ) = 0, v( n,k ) ( x, ∞ ) = v( n,k ) ( x, ∞ ) = 0 ⎪ ∂y ⎭⎪ Since governing Eq.(4) is partial differential system with an unknown parameter a(x), another condition is needed to make the solution unique. In order to make uˆ( n, k ) ( x, y ) , vˆ(n ,k ) (x, y ) and α (n, k ) (x ) satisfy the hypothesis of PSE, their variations are limited in the x-direction, alluding to normalization condition. Here selected the form as follows: ∂uˆ( n, k ) ∞ + ⎫ ∫0 uˆ(n,k ) ∂x dy = 0, for n = 2, k = 0 ⎪⎪ (6) ⎬ ⎪⎧ ∞ + ∂uˆ( n, k ) ⎪⎫ ⎪ Re ⎨ ∫ uˆ( n, k ) dy ⎬ = 0, for other modes 0 ⎪ ∂x ⎩⎪ ⎭⎪ ⎭ where, “+” denotes complex conjugate. The normalization for n = 0, k = 2 supplies two scalar equations to the determination of γ(2,0) and α. The other normalization conditions can supply one scalar equation for the unknown growth rate γ(n,k). Thus Eqs.(4)-(6) constitute nonlinear parabolic stability equations.
3 Initial Conditions and Analysis of Modes 3.1 Initial conditions The initial conditions are composed by the TS wave, mean-flow-distortion, subharmonic wave and
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the harmonics of TS wave. While the TS wave and the subharmonic wave can be defined by the results of the linear PSE. The complicated mean-flow-distortion and harmonics will be studied as follows spectively. The mean-flow-distortion (0, 0) is determined by the quadratic interaction between the TS wave and itself, and between other harmonics, thus the mean-flow-distortion will have a spatial growth rate approximately twice that of the TS wave. By denoting this exponential growth rate by α0, the calculation of the mean-flow-distortion is calculated approximately with the help of the spatial growth concept in the nonparallel basic-flow. In order to obtain an ordinary differential equation, the profiles of the mean-flow-distortion, the TS wave and the harmonics are approximatly supposed to be locally selfsimilar, e.g.,
ψ n = δ ( x) f n (η ) exp[nα n x − inωt ] n = 0,1, 2, ⋅⋅⋅
(7)
Furthermore, we can get u=
∂ψ B n =∞ n + ∑ A ⋅ f n′ ⋅ exp[α n x − inω t ] ⎫ ⎪ ∂y n =−∞
⎪ ⎪⎪ ∂ψ B n =∞ n δ ( x) y − ∑ A ⋅[ v=− fn − f n′ + ⎬ ⎪ ∂x n =−∞ 2x 2x ⎪ n≠0 ⎪ δ ( x) ⋅ f n ⋅ α n ] ⋅ exp[α n x − inω t ] ⎪⎭ n≠0
(8)
where η = y / δ ( x) , f n (η ) = φn ( x0 , y ) , and the exponent contains α n , which is the growth rate and wavenumber based on u'max. By substituting Eq.(7) into the u-component of the N-S equations, and coupling Eq.(8) and omitting the terms of order o(R–2), the following equation is yielded −
a02 1 f 0′ + a0 ( f 0′F ′ − f 0 F ′′) − (2 f 0′′′+ Ff 0′′ + R 2R F ′′f 0 ) =
∞
∑ an fn f−′′n − a− n f −′n f n′ n =−∞
(9)
n≠0
in which the terms of order R–1 on the right-hand side have been omitted, since the approximation of self-similar flow in Eq.(7) produces errors of the same order. The basic-flow streamfunction is denoted by F(η), where F is obtained by solving the Blasius equation.
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The higher-order harmonics are created by nonlinear interaction between lower-order harmonics, and it is supposed at beginning to be α n = nα1 , n > 1 (10) where, the unknowns at this point are the profiles φn and the exponent α1. Then each of these quantities (such as α1, ϕ1, ϕ2, etc.) is expanded in a Taylor series in the amplitude A. The quadratic nonlinearity of the N-S equations leads to expansions in even exponents of A, and the flow field becomes ψ ( x, y, t , A) = ψ B ( x, y ) + A2 (φ01 + A2φ02 + A4φ03 + ⋅⋅⋅) ⋅ exp(α 0 x) + A(φ10 + A2φ11 + A4φ12 + ⋅⋅⋅) exp[(α10 + A2α11 + A4α12 + ⋅⋅⋅) x − iωt ] + A2 (φ20 + A2φ21 + A4φ22 + ⋅⋅⋅) exp[2(α10 + A2α11 + A4α12 + ⋅⋅⋅) x − i2ωt ] + A3 (φ30 + A2φ31 + A4φ32 + ⋅⋅⋅) exp[3(α10 + A2α11 + A4α12 +
) x − i3ωt ]
(11)
By collecting terms multiplied by equal powers of A and arranging the governing equations in hierarchical order, and unknowns may be found out. Consistent with the definition of A based on the value of the maximum u velocity of the TS wave, the normalizations condition is chosen as follows, ∂φ ∂φ ( y ) ⎫ max( 10 ) = 10 m ⎪ ∂y ∂y ⎪ (12) ⎬ ∂φ1k ( ym ) ⎪ = 0, k = 2,3, 4, ⋅⋅⋅ ⎪⎭ ∂y in which ym is the position where the maximum of the u profile of the TS wave occurs. At orders A1, A2, ···, φ1n, α1n are obtained by solving simultaneously local partial differential equations with the normalization condition. 3.2 Analysis of modes A series of new modes will be generated in the process of C-type transition by the nonlinear interaction between the TS wave and the subharmonic wave, and the harmonic wave of TS wave. Next we analyze these new modes. The initial conditions are taken as the fundamental of the TS wave denoted by ‘T’, and the subharmonic denoted by ‘C’. With these waves being assumed to have small finite amplitude, the results of analysis are listed in Table 1. As the amplitude increases, the interaction between harmonics waves becomes so large as to gen-
erate new set of modes shown in Table 2. The small letters in the table indicate the nonlinear product giving rise to the modes. Table 1 Analysis of modes at initial condition n=0
1
2
k=0
T
k=1
C
Table 2 Analysis of new modes generated by nonlinear effects n=0 k=0 k=1 k=2
1
t2
2
c c2
3
4 c2
T ct c2
c2t
In this paper the matrix has been truncated to n = 4 and k = 2, i.e., n ∈ [−4, 4] , k ∈ [−2, 2] . The disturbance modes and the initial order given in Table 2, a set of (n, k) in the flow field has been analyzed.
4 Numerical Method 4.1 High-order spectral method and mapping In order to improve the calculation precision during solution of PSE, the high-order spectral method is used. With the Chebyshev functions Tn(y) of the first kind of Chebyshev polynomials chosen, forty Chebyshev polynomials are adopted in this paper. The approximate solution is written as following 40
40
0
0
u( n, k ) = ∑ a( n, k )mTm ( y ), v( n, k ) = ∑ b( n, k )mTm ( y ) (13)
In the calculation, the mapping is needed from the semi-infinite physical domain y ∈ [0, ∞) to calculated domain Y ∈ [−1, +1] . Here is employed algebraic mapping, because of its accuracy higher than exponential mapping[8]. It may be written in the form of Y = (by − y0 ) /(by + y0 ) . The differential operators are converted into algebraic form by aid of a spectral collocation method. Where b=1, and y0 controlling the collocation in the physical domain. Here is obtained an algebraic system, which might be solved with the help of the QR matrix algorithm. 4.2 Procedure of solving the PSE The complex velocity profiles satisfy the relationships, uˆ(n ,k ) = uˆ(n , − k ) , uˆ( n,k ) = uˆ(+− n, k ) , which are
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
based on physical characteristics of the flow-field. And, for vˆ(n ,k ) , wˆ (n,k ) , the relations can be written the same way. Thus only the modes having index n≥0, k≥0 need to be solved and stored in memory, causing a decreased number of unknowns. In using the method above-mentioned, initial conditions are given: the TS wave (2, 0), the harmonics (4, 0), the mean-flow-distortion (0, 0) and the subharmonic wave (1, 1). In the marching process of solving nonlinear PSE, firstly the solution of TS wave at each station is to be found by iterating and updating growth rate γ(2,0) and wavenumber αˆ (2,0) , and satisfying the normalization condition. Here comes the following iteration formula
(
⎡ ∞ i i Δγ ((n), k , m +1) = Re ⎢ ∫ uˆ((n),k ,m +1) + uˆ( n, k , m ) ⎣ 0
(
+
)
i uˆ((n), k , m +1) − uˆ( n, k , m ) / ( 2Δx ) dy ⎤ ⎥⎦
(
⎡ ∞ i) i) Δαˆ((2,0, = Im ⎢ ∫ uˆ((2,0, + uˆ( 2,0, m ) m +1) m +1) ⎣ 0
(
)i
)
i) − uˆ( 2,0, m ) / ( 2Δx ) dy ⎤ uˆ((2,0, m +1) ⎥⎦
)i
(14)
+
(15)
where the superscript ‘i’ denote i-th iteration, the subscript ‘m’ indicates the streamwise step number and xm = x0 + m Δ x . The ‘Re’ and ‘Im’ denote the real and the image of the parameter, respectively. The γ is growth rate, and α is wavenumber of the TS wave at this station. The number of iteration used at each step varies with the rate of convergence of γ and α. After streamwise wavenumber of the TS wave is obtained, that of other harmonics can be determined nαˆ( 2,0, m +1) αˆ( n, k , m +1) = (16) 2 Thus, the all harmonics at (m+1) station have been calculated, it is possible to march to next station by using the same method.
5 Nonlinear Evolution Analysis of C-type Instability Now the three-dimensional nonlinear C-type instability in the Blasius boundary layer is to be studied. In order to compare the solutions of the
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nonlinear PSE to the experimental data given by Saric[9], let F=106, b=0.2 in marching process of calculation, in which non-dimensional frequency F= 106ω/Re0, the non-dimensional wavenumber in the spanwise direction b=103β/R0. The velocity is nondimensionalized by using UB, and the length by δ 0 = (ν x0 / U 0 )1/ 2 , where x0 is a fixed dimensional distance from the leading edge, ν is the kinematical viscosity and R0=UBδ0/ν. The computational results of the PSE and the experimental data are graphed in the Fig.1, where the evolution curves of the amplitudes of two-dimensional wave jumps up suddenly in the vicinity of Re=688 on the experimental curves, which is ascribed to the interference of the smoke device[9], and then revert to normally at Re=700. Under this circumstance, the flow region is divided into two sections in the computation, i.e. section 1 (Re= 590-685) and section 2 (Re=700-750). The initial amplitude of the TS wave is A0=0.001 8 in section 1, and A0=0.006 4 in section 2. The initial amplitude of the subharmonics (C-Type) is B0=0.001 4. The initial values of other harmonics, which are generated by the nonlinear interactions of these two type waves, are also decided by these interactions. It is thus clear from the figure that amplitude curve of the TS wave (2, 0) has good agreement with experiment data in section 1 and section 2 after the nonlinear effects have been taken into consideration during calculation. In the process of marching and enlarging gradually along stream direction, nonlinear influences of the disturbances are also obvious. Consequently, without consideration of these influences, the results can only accord the experiment date at the position having relatively smaller Reynolds number such as in section 1, while at the position with bigger Reynolds number like in section 2 significant errors will appear[10]. The evolution curve of the amplitudes for 3-D subharmonics (1, 1) is also given in this figure. The variety of the curve is very slow at Re < 690, but at Re > 700, the variety is quick. Then growth rate is greater than that of the TS wave, and nonlinear affection is dominated, and results computed are consistent with
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experiment date. The evolution curves of modes (2, 2), (3, 1) and (4, 2) produced by nonlinear interaction are shown in Fig.2. From the figure, it follows that the nonlinear influences of these modes increases with the rise of Reynolds number, especially when Re > 700. It is the significant characteristic feature that must be paid a close attention.
Fig.1
Fig.2
Fig.3
Profile curve of mode (0,0).
Fig.4
Profile curve of mode (2,2).
Fig.5
Profile curve of mode (3,1).
Fig.6
Profile curve of mode (4,0).
Amplitude curves of modes (2,0) and (1, 1) in C-type instability.
Amplitude curve of modes (2,2), (3,1) and (4,2) in C-type instability.
The profiles of the velocity and the evolution problems related to different modes are analyzed below. In order to discuss for the convenience, the results in Figs.3-6 are obtained at the same Reynolds number (Re=638). It is obvious from the figures that the maximum value and the corresponding position for each modes generated by the nonlinear interaction of disturbance waves are obviously different, and the shape and the trend of the variation also have the remarkable dissimilarity. Fig.7 presents the profiles of mode (2, 0) at two different Reynolds numbers. By comparing the results at Re-
Liu Jixue et al. / Chinese Journal of Aeronautics 20(2007) 313-319
ynolds number (Re=736) to those at the Reynolds number (Re=638) in Fig.7, we find that the maximum value of the profile at Re=736 is bigger, and its position is farther away from the wall. Furthermore, the three-dimensional evolution of profiles in stream direction at different Reynolds number for the subharmonic (1, 1) is shown in Fig.8. Obviously, the maximum values of profiles are increased continuously, and this growth is not quick at the linear stage at the smaller Reynolds number, but when developing further downstream, the nonlinear function increases evidently, and the maximum value enlarge rapidly, and its position is farther gradually away from the wall. Obviously, the nonlinear evolution process is influenced mainly by the Reynolds number.
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harmonic wave in the C-type instability of the threedimensional disturbance is studied very effectively by PSE method and the numerical techniques developed in the paper. The different results of various modes generated by the nonlinear interaction are obtained, and Reynolds numbers are pointed out to be the decisive factor. It is proved that the presented method is in strict conformity with the experimental data. References [1]
Klebanoff P S, Tidstorm K D, Sargent L M. The three-dimensional nature of boundary layer instability. J F M 1962; 12: 1-34.
[2]
Craik A A D. Non-linear resonant instability in boundary layers. J F M 1971; 50: 393-415.
[3]
Herbert T. Secondary instability plane channel flow to subharmonic three dimensional disturbance. Physics of Fluids 1983; 26: 871-874.
[4]
Tang D B, Ma Q R, Cheng G W. Study of compressible nonparallel boundary layers. Acta Aeronautica et Astronautica Sinica 2002; 23(2): 166-169. [in Chinese]
[5]
Herbert T. On the stability of 3D boundary layers. AIAA 97-1961, 1997.
[6]
Tang D B, Xia H. Nonlinear evolution analysis of T-S disturbance wave at finite amplitude in nonparallel boundary layers. Applied Mathematics and Mechanics 2002; 23(6): 660-669.
[7]
Wang W Z, Tang D B. Studies on nonlinear stability of three-dimensional H-type disturbance. Acta Mechanica Sinica
Fig.7
2003; 19(6): 517-526.
Profile curves of TS wave. [8]
Grosch C E, Orszag S A. Numerical solution of problem in unbounded regions: coordinate transforms. Journal of Computational Physics 1997; 25: 273-296.
[9]
Saric W S, Kozlor V V, Levehenko V Y. Forced and unforced subharmonic resonance in boundary layer transition. AIAA-840007, 1984.
[10]
Lu C G, Zhao G F. Theoretical analysis of C-type instability in the Blassius boundary layer. Aerodynamica Sinica 1995; 13(3): 329-333. [in Chinese]
Biography: Liu Jixue Fig.8
Profile curves of mode (1,1) at different Reynolds numbers change with stream direction.
6 Conclusions Nonlinear evolution process of TS wave and
Born in 1979, he received B.S.
and M.S. from Nanjing Aeronautical Institute in 2003 and 2006 respectively, and then became an engineer in Chengdu aircraft design institute.