Fluid Dynamics, Vol. 37, No. 1, 2002, pp. 9–20. Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2002, pp. 13–25. Original Russian Text Copyright 2002 by Ustinov.
Stability of the Flow in a Streaky Structure and the Development of Perturbations Generated by a Point Source Inside It M. V. Ustinov Received November 24, 2000
Abstract — The flow stability in a boundary layer with an inhomogeneous spanwise-periodic velocity profile modeling the streaky structure that develops at a high level of turbulence of the incident flow is analyzed in the three-dimensional formulation for perturbations with an arbitrary transverse period. It is shown that in the presence of inhomogeneity the dispersion relation for the Tollmien-Schlichting waves is split into two branches periodic in the transverse wave number, which correspond to symmetric and antisymmetric modes. The solution for the packet of inhomogeneous-flow modes generated by localized time-periodic fluid injection/ejection is found. The shape of this packet corresponds qualitatively to the shape of the Tollmien-Schlichting wave packet, but the fine perturbation structure inside it is sharply different.
At a high level of turbulence of the incident flow (0.1 < εT < 10%) laminar-turbulent transition in the boundary layer begins with the growth of low-frequency perturbations [1]. Flow visualization [2] shows that these perturbations form a streaky structure, alternating long streaks with increased and decreased velocity which appear and disappear randomly with a relatively low frequency. Downstream, against the background of this structure, high-frequency perturbations grow rapidly leading to turbulization of the flow. So far, only the first stage of transition at high levels of turbulence, the formation of a streaky structure, has had a satisfactory theoretical description. The development of high-frequency perturbations of this structure remains obscure in many respects despite the numerous experimental studies of this problem. In [3] it was shown that in a boundary layer with a streaky structure artificially created Tollmien-Schlichting waves can develop and accelerate the transition process. On the other hand, in a boundary layer with artificially created inhomogeneity modeling the streaky structure, the development of higher-frequency perturbations fundamentally different from Tollmien-Schlichting waves has been observed [4]. Similar perturbations, associated with spanwise inflectional instability of the velocity profile, have been found as a result of a theoretical analysis of the stability of a flow with streamwise vortices [5]. The flow stability in a streaky structure was investigated theoretically in the temporal approach in [6]. In [6], in contrast to [5], the base flow did not contain streamwise vortices. In [6], it was shown that in the streaky structure unstable perturbations of two types with comparable growth increments exist, namely, symmetric modes similar to Tollmien-Schlichting waves and antisymmetric modes due to the spanwise instability of the velocity profile. This study continues [6] in two directions: analysis of the stability of the streaky structure in the three-dimensional formulation for perturbations with an arbitrary transverse period and investigation of the development of packets of these perturbations generated by a localized source. 1.
FORMULATION OF THE PROBLEM
Let us consider the flow of a viscous incompressible fluid with the kinematic viscosity ν in the boundary layer on a flat plate. We locate the origin of the Cartesian coordinate system on the plane surface at a distance L from the leading edge. We direct the X , Y , and Z coordinate axes streamwise, normally to the plate surface, and in the transverse direction, respectively. The scales for the coordinates and velocity are the boundary-layer thickness δ = ν L/u∞ and the velocity of the incident flow u∞ . We will assume that in the 0015–4628/02/3701–0009$27.00 2002 Plenum Publishing Corporation
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boundary layer there is a steady-state span-periodic inhomogeneity of the longitudinal velocity component created by an external effect concentrated far away upstream from the coordinate origin. Taking a boundary layer homogeneous in x, which corresponds to its state at the coordinate origin, for the basic flow, we will consider the development of infinitely small perturbations generated by time-periodic injection/ejection through a localized site on the plate surface. For this purpose, we represent the velocity field in the form: V(x, y, z, t) = Vb (y, z) + ε V p (x, y, z)e−iω t Vb = {U0 (z) + ub (z) cos β0 y, 0, 0},
(1.1)
V p = {u, v, w}
w(x, y, 0) = f (x, y)
(1.2)
where the first term represents the base flow and the second describes the small perturbations produced by injection/ejection distributed according to law (1.2) in which f is a finite function. The other components of the perturbation velocity on the wall satisfy the non-slip condition. The perturbations are also assumed to be damped at a large distance from the wall and upstream from the injection/ejection site: u, v(x, y, 0) = 0,
u, v, w(x, y, ∞) = u, v, w(−∞, y, z) = 0
(1.3)
For describing the perturbation development, we will use a system of equations for the vertical velocity and vorticity components which is equivalent to the Navier-Stokes equations. Linearizing this system about the base flow yields the equations for the perturbations 2 ∂ Ub ∆ ∂ ∂ 2 − iω + ∆− − + β0 ub cos β0 y Ub 2 ∂x R ∂z ∂x ∂2 ∂ ∂v w + 2β0 sin β0 y u =0 2β0 ub sin β0 y ∂ x∂ y ∂z b ∂x ∂ Ub ∂ ∂ ub ∆ ∂ ∂ + β0 sin β0 y ub − w − β02ub cos β0 y v = 0 η+ Ub − iω + ∂x R ∂z ∂y ∂z ∂z
(1.4)
∂u ∂v ∂w + + =0 ∂x ∂y ∂z η=
∂u ∂v − ; ∂y ∂x
∆=
∂2 ∂2 ∂2 + + 2 2 2 ∂x ∂y ∂z
where R = u∞ δ /ν is the Reynolds number and Ub = U0 + ub cos β0 y is the streamwise component of the base-flow velocity. We will solve these equations using a Fourier transformation in the variables x and y and associating the arbitrary function g(x, y) with its Fourier transform g∗ (k, β ) calculated from the formula 1 g∗ (k, β ) = F[g] = 2 4π
+∞
+∞
dx −∞
dy g(x, y)e−i(kx + β y)
−∞
If the perturbations are not damped as x → +∞, this transformation is not lawful and must be regarded as a formal procedure whose significance will become clear later. Applying the Fourier transformation to Eqs. (1.4) and boundary conditions (1.2) and (1.3) with account for the equalities F[g cos β0 y] =1 /2 [g− + g+ ], FLUID DYNAMICS
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g± = g∗ (k, β ±β0 )
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STABILITY OF THE FLOW IN A STREAKY STRUCTURE
we obtain the following boundary-value problem for ordinary differential equations: ∆∗ ik i(kU0 − ω ) − ∆∗ w∗ − ikU0 w∗ + [c− ub w− + (c− − 1)ub w− − (ub + γ 2 ub )w− + R 2 d− (ub η− ) + c+ ub w+ + (c+ − 1)ub w+ − (ub + γ 2 ub )w+ + d+ (ub η+ ) ] = 0 ∆∗ i η∗ + iβ U0 w∗ + [e− (ub w− ) + h− ub η− + e+ (ub w+ ) + h+ ub η+ ] = 0 i(kU0 − ω ) − R 2 w∗ (0) = η∗ (0) = w∗ (∞) = η∗ (∞) = 0
w∗ (0) = f∗ (k, β ), ∆∗ =
d2 − γ 2, dz2
e± = ±
(1.5)
c± =
β0 2 (γ ± β β0 ), γ±2
1 2 (γ ± β02 ), γ±2
h± =
d± = ±
k 2 (γ ± 2β β0 ), γ±2
2kβ0 γ±2
γ 2 = k2 + β 2 ,
γ±2 = k2 + (β ± β0 )2
where a prime denotes the derivative with respect to z. These equations contain, in addition to the Fourier transforms of the vertical velocity and vorticity w∗ and η∗ at the point k, β , their Fourier transforms at the displaced points k, β ± β0 denoted by w± and η± . In order to find their solution, we will first assume the Fourier transform of the injection/ejection shape f∗ to be nonzero only within a neighborhood of the arbitrary wave number β∗ that is small as compared with β0 and will be denoted by O(β∗ ). It is then obvious from the form of Eqs. (1.5) that their solutions w∗ and η∗ are nonzero in both the neighborhood of β∗ and the same neighborhood of the points β∗n = β∗ ± nβ0 , where n = 0, 1, 2 . . . . For convenience, we will represent this solution as a sum of finite functions wn and ηn , each of which describes the solution in the neighborhood of the n-point β∗n : w ∗ , η∗ =
N
∑
wn , ηn (k, β )
(1.6)
n=−N
wn , ηn (k, β ) = w∗ , η∗ (k, β + nβ0 ), wn , ηn (k, β ) = 0,
β ∈ O(β∗ )
β∈ / O(β∗ )
Considering β in the neighborhood of β∗ , it is easy to see that in this neighborhood w∗ , η∗ (β ) = wn , ηn (β ),
w± , η± (β ) = w∗ , η∗ (β ±β0 ) = wn±1 , ηn±1 (β )
and (1.5) reduces to the following system of equations for wn , ηn , wn±1 , and ηn±1 : L(k, βn )wn + L+ (k, βn )wn+1 + L− (k, βn )wn−1 + N +(k, βn )ηn+1 + N −(k, βn )ηn−1 = 0
(1.7)
M(k, βn )ηn + R+ (k, βn )wn+1 + R− (k, βn )wn−1 + M +(k, βn )ηn+1 + M −(k, βn )ηn−1 = 0 w0 (0) = f∗ (k, β ),
wn (0) = 0,
n = 0
wn (0) = ηn (0) = wn (∞) = ηn (∞) = 0 where βn = β + nβ0 and L, L± , M, M ± , and R± are linear differential operators whose form is obvious from comparison with (1.5). Since in this reasoning n is arbitrary, relations (1.7) hold for all n. Writing these equations for all n = 0, ±1, ±2, . . . , ±N and setting w±(N+1) , η±(N+1) ≡ 0, we obtain a closed system of 2 × (2N + 1) equations for 2 × (2N + 1) unknown functions wn and ηn . The Fourier transforms of the streamwise and transverse velocity components u∗ and v∗ are also sought in the form of sums similar to FLUID DYNAMICS
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(1.6), whose terms un and vn are expressed in terms of wn and ηn from the transformed continuity equation and the definition of the vertical vorticity: u, v(k, β ) =
N
∑
un , vn (k, β )
n=−N
un = (ikwn − iβn ηn )/γn2 ,
vn = (iβn wn + ikηn )/γn2 ,
γn2 = k2 + βn2
(1.8)
Let us now consider an injection/ejection distribution of general form, whose Fourier transform is nonzero on a fairly broad interval of wave numbers β . Dividing this interval into a certain number of subintervals, narrow as compared with β0 , we represent f∗ as a sum of finite functions nonzero on these subintervals. For each of these functions the solution has the form (1.6), (1.8) and can be found from (1.7). Due to the linearity, the solution of the problem for the total Fourier injection/ejection transform f∗ is the sum of the solutions for its constituent finite functions. Therefore, the total solution also has the form (1.6), (1.8) and can be found from the boundary value problem (1.7). We note that, in contrast to the case of a narrow finite function f ∗ considered above, the terms wn and ηn in (1.6) are nonzero all across the broad interval β on which f∗ = 0. Now the second and third equalities (1.6), which define wn and ηn , lose their meaning and for all β the Fourier transforms of the vertical velocity and vorticity components are sums of at least a few terms of the first sum (1.6). In the case of a “broad” Fourier transform f∗ (β ), the functions wn and ηn (β ) must be regarded as simply solutions of system (1.7). In what follows, we will also consider problem (1.7) for complex k. For homogeneous boundary conditions on the plate surface, we obtain from (1.7) an eigenvalue problem which describes the stability of the periodic flow in the boundary layer in the three-dimensional formulation. Its solutions k j (ω , β ) correspond to unstable modes of the form N
Vp =
∑
Vn j (z)ei(k j x+βn y−ω t) ,
Vn j = {un j , vn j , wn j }
(1.9)
n=−N
where un j and vn j can be expressed by means of (1.8) in terms of the eigenfunctions wn j and ηn j corresponding to the eigenvalue k j . At the points k = k j (ω , β ) the solutions of the nonhomogeneous system (1.7) wn and ηn , as well as un and vn , have poles. At these poles, the residues of the solution of the nonhomogeneous system are proportional to the eigenfunctions corresponding to k j . When the transformation β → β + mβ0 ; wn , ηn (β + mβ0 ) → wn+m , ηn+m (β ) is applied, the homogeneous system (1.7) transforms into itself and the nonhomogeneous one into a similar system with modified boundary conditions on the wall wm (β ) = f∗ (β + mβ ), wn (β ) = 0, n = m. This means that the dispersion relation for unstable modes of the periodic flow k = k j (ω , β ) is a periodic function β with the period β0 and at points a whole number of periods apart the eigenfunctions are linked by the relation k j (ω , β + mβ0 ) = k j (ω , β ),
{un , vn , wn }(β + mβ0 ) = {un+m , vn+m , wn+m }(β )
(1.10)
Substituting this equality in (1.9) shows that at these points the perturbations corresponding to the eigenfunctions are identical, that is, modes with β a period apart describe the same phenomenon. An important consequence of (1.10) is the fact that for β a whole number of periods apart the poles of solutions of the nonhomogeneous system (1.7) coincide and the residues at the poles are linked by the relation res {un , vn , wn }(β + mβ0 , z) = cm (β ) res {un+m , vn+m , wn+m }(β , z)
k=k j
k=k j
(1.11)
where the coefficient of proportionality cm depends on m and β but not on n and z. This relation can be obtained directly from (1.10) and from the proportionality of residues of solutions of the nonhomogeneous system to the eigenfunctions of the homogeneous system. FLUID DYNAMICS
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STABILITY OF THE FLOW IN A STREAKY STRUCTURE
If the inhomogeneous flow is stable to perturbations of a fixed frequency ω for all β , the perturbations are damped as |x, y| → ∞ and are determined by the inverse Fourier transformation. In what follows, we will restrict ourselves to finding the perturbations of the streamwise velocity component which in the stable flow have the form: N
u(x, y, z, t) =
∑
+∞
unf (β ,
iβ y
x)e
dy,
unf (β ,
+∞
un (k, β )eikx dk
x) =
n=−N−∞
(1.12)
−∞
We will analyze the asymptotics of the solution obtained as |x| → ∞, for which purpose we will first consider its Fourier transform in β , unf (β , x), for large x. To do this, using the Cauchy theorem, we rewrite the second integral in (1.12) in the form: unf (β ,
x) = − lim
un (k, β )eikx dk + 2π i ∑ res un (k, β )eik j x
r→∞
k=k j
cr
Here, cr is a semicircle of radius r lying in the upper half-plane and the summation in the second term is performed over the poles located between cr and the real axis. If the flow is stable, for example, when the inhomogeneity amplitude is small, all the poles lie above the real axis and both terms are damped rapidly as x → ±∞. With increase in the inhomogeneity amplitude, one of the poles located at the point k0 approaches the real axis. As x → +∞, the solution gradually comes to be determined by the contribution of this pole and is damped slowly as exp(− Im(k0 (x))). With subsequent increase in the inhomogeneity amplitude, the flow becomes unstable, the pole crosses the real axis, and the term it determines disappears. At this point, the solution changes suddenly, being damped as x → +∞ instead of growing due to destabilization. In order to obtain a physically reasonable solution, it is necessary, in accordance with the principle formulated in [7], to add to the expression for unf (1.12) at Im(k0 ) < 0 the term determined by the residue at the point k0 : unf (β ,
+∞
un (k, β )eikx dk + unr (β )eik0 x ,
x) =
unr (β ) = 2π i res un (k, β ) k=k0
−∞
The solution supplemented in this way becomes continuous as the pole crosses the real axis and grows as x → ∞ in the presence of unstable modes. As x → +∞, the integral entering into the expression for unf remains finite and can be neglected as compared with the residue-related term which grows exponentially. Taking this into account, from (1.12) we obtain the following expression for the velocity perturbations at large x: N
u=
∑
+∞
n=−N−∞
unr (β )ei(k0 (β ) + αβ )x d β ,
α=
y x
(1.13)
Similar expressions for the unstable boundary-layer perturbations generated by localized surface vibration were obtained in [8, 9]. These expressions were derived in detail and justified in [8], which dealt specifically with this question. It follows from the numerical results reproduced below that system (1.7) generally has two poles in the lower half-plane which correspond to two types of unstable modes of the inhomogeneous flow. The growth increments have maxima with respect to β at β = ±mβ0 for modes of the first type and at β = β0 /2 ± mβ0 for those of the second (m = 1, 2 . . . ). At the maximum points, the streamwise velocity components of these modes are distributed over y symmetrically and antisymmetrically, respectively. We will call these modes symmetric and antisymmetric. For α and ω values at which unstable modes of both types exist, the velocity perturbations are sums of two terms of the (1.11) type, one of which takes into account the contribution of the symmetric and the other that of the antisymmetric modes. FLUID DYNAMICS
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We will first restrict ourselves to the term corresponding to the symmetric modes. For these the real part of the exponent in (1.13) has maxima at the points βm = ±mβ0 (m = 1, 2, . . . ) whose neighborhoods make main contribution to the integral for u as x → ∞. Estimating the contribution of each maximum to the integral by the saddle-point method, we obtain the relation
M N 2π u= ∑ ∑ unr (βs + mβ0) exp{i[k0 (βs ) + (βs + mβ0)α ]x + iθ } |K0 (βs )|x m=−M n=−N Here, βs is the saddle-point nearest to β = 0 of the function ψ (β ) = i(k0 (β ) + αβ ) considered for complex β , which is found from the condition ψ (βs ) = 0, and θ is the angle between the line Im(ψ ) = const passing through βs and the real axis. This expression has been derived with account for the periodicity of the dispersion relation for spanwise-periodic flow, from which it follows that the saddle-points corresponding to the maxima at βm can be obtained by displacing βs by mβ0 along the real axis and K0 and θ are the same at all saddle-points. Using relation (1.11), we rewrite the expression for the perturbations in the more convenient form: u = A(α , x)[Φ(α , y) exp{i(ksr (α )x + βsr (α )y}], Φ(α , y) =
1 A(α , x) = √ eψr (α )x x
N 2π C(βs ) ∑ unr (βs (α ))einβ0 y eiθ (α ) , |K0 (βs )| n=−N
C(βs ) =
M
∑
(1.14)
cm (βs )
m=−M
where ksr , βsr , and ψr are the real parts of k0 , β , and ψ at the saddle-point and cm (βs ) is the proportionality coefficient from (1.11) in the residue k0 . This expression is the product of the almost x- and y-periodic function in square brackets, which describes the flow corresponding to the unstable mode of inhomogeneous flow, and the slowly-changing amplitude factor A(α , x). The curve on which A(α , x) reaches a certain fixed value can be interpreted as the boundary of the disturbed region. With logarithmic accuracy it coincides with the curve of constant value of the exponent in the expression for A x = const/ψr (y/x)
(1.15)
The shape of the disturbed region does not depend on the injection/ejection distribution f (x, y) and is determined exclusively by the dispersion relation for unstable modes of the inhomogeneous flow. Formulas (1.14) and (1.15) are also valid for the antisymmetric modes. The only difference lies in the fact that βs is now the saddle-point nearest to β0 /2 and k0 (β ) is the position of the pole corresponding to the antisymmetric mode. For numerical solution, problem (1.7) was reduced to a system of differential equations with right sides and homogeneous boundary conditions. This system was discretized using a collocation method similar to that described in [10]. As collocation points, the zeros of the Jacobi polynomial PQ(1,1) (z) were used. The number of nodes Q was 51, and the number of “harmonics” in the representation for perturbations (1.6) N was 11. As shown by subsequent calculations, this number of degrees of freedom is sufficient for finding the imaginary part of the eigenvalue k0 with an accuracy of up to 2–3% of its maximum value. After discretization, the problem was reduced to a system of (2Q − 1) × (2N + 1) linear equations for the values of the unknown functions at the collocation nodes. Owing to the linearity of the system of equations (1.7), the residues of all the functions entering into this system are proportional to the Fourier injection/ejection transform at the point k0 : f∗ (k0 , β ). Therefore, the residues were calculated for f∗ (k0 , β ) ≡ 1 and then multiplied by the Fourier injection/ejection transform. To do this, we calculated the function g(x) = |u0 (k)|−1 and sought by Newton’s method the point k0 at which this function is equal to zero. Simultaneously, we calculated the derivative g (k) which was used for finding the residue res u0 =
k=k0
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1 g (k0 )
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STABILITY OF THE FLOW IN A STREAKY STRUCTURE
Fig. 1. Profiles of the constant U0 (z) (1) and periodic ub (z) (2) components of the base flow
The saddle-point βs was calculated by Newton’s method as the zero of the function ψ (β ). The coefficients of proportionality cm were found directly by calculating the solution residues at the point k0 (βs + mβ0 ). The residues at the displaced points βs + mβ0 decreased rapidly with increase in m; therefore, for a good approximation of the solution, M = 4 proved to be sufficient. 2.
CALCULATION RESULTS
The aim of our study was to analyze the evolution of the perturbations in a boundary layer with a streaky structure generated by external turbulence. Therefore, the profiles of the constant U0 (z) and periodic ub (z) components of the base flow were chosen to satisfy the condition of maximum similarity to the profiles of the average velocity and the low-frequency pulsations in such a boundary layer. In [1], no average-velocity deviations from the Blasius solution were observed; therefore, we used this solution as U0 (z): U0 (z) = f (z),
f +
1 f f = 0, 2
f (0) = f (0) = 0,
f (∞) = 1
(2.1)
The velocity pulsation profile in experiments [1, 2] can be very well described by the analytic solution obtained in [11], which was used to assign the inhomogeneity profile: ub (z) = a
η f max(η f )
(2.2)
z
where f is the Blasius function from (2.1) and a is the inhomogeneity amplitude. This solution also almost coincides with the inhomogeneity profiles generated in the boundary layer by spanwise [12] and vertical [13] vortices in the incident flow. The profiles of the constant (2.1) and periodic (2.2) components of the base flow are shown in Fig. 1. The subsequent calculations of the perturbation evolution were performed for β0 = 0.6 and R = 600. The wave number β0 was based on the requirement that the inhomogeneity period λ = 2π /β0 should fit the experimental data [1]. The smallness of the Reynolds number chosen is attributable to the early laminar-turbulent transition at high levels of turbulence. Before finding the perturbation packets due to a localized source, we analyzed the dispersion relation for the symmetric and antisymmetric modes. The results obtained are presented in Fig. 2 in the form of dependences of the real and imaginary parts of the eigenvalue k0 on the transverse wave number β . The dispersion relation is periodic in β ; therefore, only one period, from 0 to β0 , is shown. The dependences k0 (β ) for the Tollmien-Schlichting waves (at a = 0) and a weakly inhomogeneous flow at a = 0.05 show how the aperiodic dispersion relation for a homogeneous boundary layer transforms smoothly into a periodic FLUID DYNAMICS
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Fig. 2. Real (a) and imaginary (b) parts of the eigenvalue k0 vs. β at ω = 0.05 for different values of the inhomogeneity amplitude a: continuous curve — Tollmien-Schlichting waves (a = 0); dotted curve — a = 0.05; broken curve — a = 0.2, and chain curve — a = 0.3. 1 — symmetric and 2 — antisymmetric modes
relation. It turns out that in the presence of an arbitrarily small inhomogeneity the dispersion relation splits into two branches. The first of these, numbered 1, is similar to the dispersion relation for the homogeneous flow on the first half of the period 0 < β < β0 /2, while the second (2) is similar to that on the second half β0 /2 < β < β0 . On the other half-periods these branches are similar to the dispersion equation for the Tollmien-Schlichting waves displaced by a period (plotted as a heavy broken line). At β = 0 and β = β0 /2, the first-branch modes correspond to perturbations with symmetric velocity distributions u(y) and the second-branch modes to asymmetric perturbations. Therefore, as mentioned above, we will call the first-branch modes symmetric and the second-branch modes antisymmetric. It becomes clear from the way the dispersion relation splits into symmetric and antisymmetric modes and the retention of the general form of the dependences Im(k0 )(β ) with increase in the inhomogeneity amplitude why the maxima of the growth increments are located at β = ±nβ0 for the symmetric modes and at β = β0 /2 ± nβ0 for the antisymmetric modes. These β values correspond to the main-period and subharmonic modes considered earlier in [6]. The conclusion in [6] that the most rapidly growing perturbations are the main-period modes and the most unstable antisymmetric perturbations are the subharmonic modes is in complete agreement with our results. From Fig. 2, the inverse conclusion also follows that the most slowly growing symmetric modes are subharmonic and the most stable antisymmetric modes are the main-period ones. For the symmetric modes, the β -dependences of the real part of k0 vary relatively weakly with increase in the inhomogeneity amplitude and, especially near the maxima of the growth increments, are almost identical to the dispersion relation for the Tollmien-Schlichting waves. This corresponds to the conclusion reached in FLUID DYNAMICS
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Fig. 3. Growth increments maximal over β , σ = − Im(k0 ), vs. frequency ω : 1–3 — symmetric modes at a = 0.1, 0.2, and 0.3; 4 — antisymmetric modes at a = 0.3; 5 — Tollmien-Schlichting waves
[6] that the phase velocity of the symmetric modes is close to the velocity of the Tollmien-Schlichting wave. For the antisymmetric modes, the real part of k0 decreases considerably with increase in the inhomogeneity amplitude, which at a ≥ 0.2 leads to an increased phase velocity of these modes c = ω /k0 0.6, as also mentioned in [6]. The frequency dependences of the growth increments, maximal in β , of the symmetric and antisymmetric modes σ = − Im(k0 ) are shown in Fig. 3 for different inhomogeneity amplitudes. Clearly, at a moderate inhomogeneity amplitude a = 0.1 the growth increments of the symmetric modes are almost the same as those of the Tollmien-Schlichting waves. With further increase in the inhomogeneity amplitude, the growth increments of these modes increase considerably and the range of unstable frequencies is extended. Moreover, at the greatest amplitude analyzed a = 0.3, the maximum growth increment is displaced in the direction of greater frequencies. The results obtained make it possible to explain the constancy of the TollmienSchlichting wave characteristics in the boundary layer at high levels of turbulence, which was discovered in [3], where the amplitude of the low-frequency pulsations (inhomogeneity) in the boundary layer was equal to ∼ 10%. On the other hand, at high levels of turbulence in the boundary layer, high-frequency pulsations grow when the amplitude of the low-frequency pulsations reaches 20–30%, which roughly corresponds to the minimum inhomogeneity amplitude leading to a substantial change in the stability characteristics of the boundary layer. The dependence σ (ω ) for antisymmetric modes is shown in Fig. 3 for a = 0.3 only, since for the other inhomogeneity amplitudes studied, a = 0.1 and 0.2, these modes turned out to be damped. Even for such a large inhomogeneity amplitude, the growth increments of the antisymmetric modes proved to be smaller than the growth increments of the symmetric perturbations. The disagreement with [6], where at a = 0.3 the opposite situation was observed, is attributable to the difference in the inhomogeneity period. We also note the much greater frequency of the growth increment maximum for the antisymmetric modes, ω 0.11, as compared with the frequency of the most rapidly growing Tollmien-Schlichting waves, ω 0.05, and the symmetric modes for a = 0.3, ω 0.07. The greater frequency of the antisymmetric modes as compared with the symmetric ones corresponds to the experimental data [4]. In a realistic boundary layer, the parameters of the perturbations change gradually with their downstream propagation due to changes in the base flow. In order to estimate this effect, we will assume the inhomogeneity period to be constant along the longitudinal coordinate. Then, due to the time homogeneity and FLUID DYNAMICS
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Fig. 4. Growth increments maximal over β , σ = − Im(k0 ), vs. x: 1–2 — symmetric modes at ω = 0.05 and a = 0.2 and 0.3; 3 — antisymmetric mode at ω = 0.1 and a = 0.3; 4 — Tollmien-Schlichting wave at ω = 0.05
Fig. 5. Shape of the undisturbed region for perturbations generated by a point source. Notations as in Fig. 3
stepwise periodicity of the base flow, the time and transverse perturbation periods remain constant over x and, hence, ω and β also remain constant. The perturbation parameters change due to the change in the velocity profiles of the average and periodic components of the base flow. The latter are assumed to be stationary in the coordinates fitted to the local thickness of the boundary layer, that is, are determined by (2.1) and (2.2), where f is a function of the variable z∗ = z/ 1 + x/R. The dependences of the growth increments of the Tollmien-Schlichting waves and the unstable modes of the inhomogeneous flow on the longitudinal coordinate x obtained on these assumptions are shown in Fig. 4. From these it can be seen that the inhomogeneity leads not only to an increase in the growth increments and the range of unstable frequencies but also to an elongation of the region on which perturbations of a fixed frequency grow. This effect is strongest for the antisymmetric perturbations which continue to grow at an almost constant rate with increase in x. In spite of the smaller growth increments at x = 0, these modes may be even more dangerous than the symmetric ones due to the longer growth interval and the broader range of unstable frequencies. The dependences σ (x) also make it possible to estimate the limits of applicability of the expressions for the perturbations generated by a point source (1.13) and (1.14) obtained in the plane-parallel approximation. The latter assumes the eigenvalue k0 to be independent of x and to be a function of the transverse wave number β only. For the real part of k0 which determines the perturbation wavelength, this assumption is satisfied with an error of ∼ 1% (the dependences Re(k0 )(x) are not reproduced to save space). As can be seen from Fig. 4, the imaginary part of the eigenvalue Im(k0 ) = −σ depends strongly on x. However, we can expect the expressions for point-source perturbations (1.13) and (1.14) to give qualitatively correct results when the change in σ with increase in x is small as compared with its dependence on β . Since the growth increments vary with respect to β by their own magnitude, the plane-parallel approximation can be considered as qualitatively correct when σ varies with respect to x by 20–30%. It can be seen from FLUID DYNAMICS
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STABILITY OF THE FLOW IN A STREAKY STRUCTURE
Fig. 6. Distribution of the amplitudes of longitudinal velocity component pulsations in the (x, y) plane for a packet of symmetric modes at ω = 0.05 and a = 0.3. The isoline step is equal to 4
Fig. 4 that for perturbations (symmetric and antisymmetric) of a strongly inhomogeneous flow at a = 0.3 this condition holds up to x = 400–500 and for symmetric modes at a = 0.2 and Tollmien-Schlichting waves up to x = 250–300. Since at a ≤ 0.3 the symmetric modes proved to grow the most rapidly, we mainly investigated packets of such modes generated by localized injection/ejection at a frequency ω = 0.05. The asymptotic shape of these packets calculated from formula (1.15) for different inhomogeneity amplitudes is shown in Fig. 5. It does not differ qualitatively from the shape of a packet of Tollmien-Schlichting waves, except for a certain broadening at the maximum inhomogeneity amplitude a = 0.3. The only calculated packet of antisymmetric modes for a = 0.3 and ω = 0.1 proved to be considerably narrower than the packets of symmetric modes even for a smaller amplitude a = 0.2. The major difference between the packets of perturbations of the inhomogeneous flow and the packets of Tollmien-Schlichting waves lies in their small-scale structure. The latter is shown in Fig. 6, where the isolines of modulus of the longitudinal velocity component perturbations |u| in the (x, y) plane, calculated from the exact formula (1.14) for symmetric modes at a = 0.3 and ω = 0.05, are depicted. These results were obtained for a Fourier transform f∗ (k, β ) = 1 corresponding to injection/ejection distributed in accordance with the Dirac delta function f (x, y) = 2πδ (x)δ (y). Physically, this distribution corresponds to a very narrow and short, as compared with the wavelength and the transverse perturbation period, injection/ejection zone. In the inhomogeneous flow, in contrast to the smooth distribution of perturbation amplitudes in the Tollmien-Schlichting wave packet (see [8, 9]), the pulsations are concentrated in narrow streaks corresponding to the lower-velocity streaks in the base flow. This behavior of the perturbations is fairly obvious and is in agreement with the experimental data [4] and the distribution u(y) for symmetric modes obtained in [6]. An unexpected result is the gradual transformation of the velocity distribution with a single maximum coinciding with the base-flow velocity minimum into a distribution with two maxima on different sides of the minimum with movement from the center toward the periphery of the packet. This change in pulsation shape can be explained by the fact that at the center of the packet, in accordance with (1.14), the perturbations are determined by the mode with β = 0, which has a maximum at the point of velocity minimum, and at the periphery by the mode with a complex β = βs , which presumably has a velocity distribution with two maxima. Summary. The stability of the flow in a boundary layer with spanwise-periodic inhomogeneity of the velocity profile modeling a streaky structure is analyzed. With the appearance of inhomogeneity of arbitrarily small amplitude, the dispersion relation for the Tollmien-Schlichting waves splits into two branches periodic in β , which correspond to symmetric and antisymmetric modes. The symmetric modes grow most rapidly, FLUID DYNAMICS
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but the antisymmetric perturbations are unstable on a wider frequency range and for a fixed frequency grow over a longer distance downstream. The solution for a packet of inhomogeneous-flow modes generated by time-periodic localized fluid injection/ejection is found. In contrast to the smooth distribution of amplitudes in a Tollmien-Schlichting wave packet, the inhomogeneous flow perturbations are concentrated in narrow streaks associated with a lower velocity of the base flow. In each streak at the center of the packet there is a single pulsation maximum coinciding with the base-flow velocity minimum and at the periphery two maxima on opposite sides of a minimum. The work was carried out with the financial support of the Russian Foundation for Basic Research (project Nos. 01-01-00614 and 00-15-96069). REFERENCES 1. K. J. A. Westin, A. V. Boiko, B. G. Klingmann, V. V. Kozlov, and P. H. Alfredsson, “Experiments in a boundary layer subjected to free-stream turbulence. Pt. 1: Boundary layer structure and receptivity,” J. Fluid Mech., 281, 193–218 (1994). 2. P. H. Alfredsson and M. Matsubara, “Streaky structures in transition,” Transitional Boundary Layers in Aeronautics (R. A. W. M. Henkes and J. L. van Ingen, eds.), Elsevier, Amsterdam, 373–386 (1996). 3. A. V. Boiko, K. J. A. Westin, B. G. Klingmann, V. V. Kozlov, and P. H. Alfredsson, “Experiments in a boundary layer subjected to free-stream turbulence. Pt. 2: The role of TS-waves in the transition process,” J. Fluid Mech., 281, 219–246 (1994). 4. A. A. Bakchinov, G. R. Grek, B. G. Klingmann, and V. V. Kozlov, “Transition experiments in a boundary layer with embedded streamwise vortices,” Phys. Fluids, 7, No. 4, 820–832 (1995). 5. Y. Xiuyang and J. T. C. Liu, “The secondary instability in Goertler flow,” Phys. Fluids A, 3, No. 8, 1845–1847 (1991). 6. M. V. Ustinov, “Stability of a span-inhomogeneous flow in a boundary layer,” Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 6, 54–63 (1998). 7. E. D. Terentyev, “Linear problem of a vibrator oscillating harmonically at supercritical frequencies in a subsonic boundary layer,” Prikl. Mat. Mekh., 48, No. 2, 264–272 (1984). 8. O. S. Ryzhov and I. V. Savenkov, “Spatial perturbations introduced by a harmonic oscillator into the boundary layer on a plate,” Zh. Vychisl. Mat. Mat. Fiz., 28, No. 4, 591–602 (1988). 9. S. V. Manuilovich, “On the receptivity of flow in a boundary layer to vibration of a localized surface site,” Dokl. Akad. Nauk SSSR, 305, No. 3, 563–566 (1989). 10. B. L. Rozhdestvenskii and M. I. Stoinov, Algorithms for Solving the Navier-Stokes Equations Having Analogs to the Mass, Momentum and Energy Conservation Laws, M. V. Keldysh Institute of Theoretical and Applied Mechanics, Novosibirsk, Preprint No. 119 (1987). 11. S. C. Crow, “The spanwise perturbation of two-dimensional boundary layers,” J. Fluid. Mech., 24, Pt. 1, 153–164 (1966). 12. P. Andersson, M. Berggren, and D. S. Henningson, “Optimal disturbances and bypass transition in boundary layers,” Phys. Fluids, 11, No. 1, 134–150 (1999). 13. M. V. Ustinov, “Receptivity of the boundary layer on a flat plate with blunt leading edge to a steady-state inhomogeneity of the incident flow,” Prikl. Mekh. Tekhn. Fiz., No. 4, 93–100 (2000).
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