The onset of Görtler vortices in laminar boundary layer flow over a slightly concave wall

European Journal of Mechanics B/Fluids 29 (2010) 407–414

Contents lists available at ScienceDirect

European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu

The onset of Görtler vortices in laminar boundary layer flow over a slightly concave wall Min Chan Kim a,∗ , Chang Kyun Choi b , Do-Young Yoon c a

Department of Chemical Engineering, Jeju National University, Jeju 690-756, Republic of Korea

b

School of Chemical and Biological Engineering, Seoul National University, Seoul 151-744, Republic of Korea

c

Department of Chemical Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea

article

info

Article history: Received 12 November 2009 Received in revised form 12 July 2010 Accepted 16 July 2010 Available online 24 July 2010 Keywords: Görtler vortex Linear stability theory Propagation theory Relative instability

abstract The onset of convective instability in the laminar boundary layer over the slightly curved wall is analyzed theoretically and compared with the existing experimental data. A new set of stability equations are derived by the propagation theory considering the relative instability under the linear stability theory. In this analysis the disturbances are assumed to have the form of longitudinal vortices and also to grow themselves in streamwise direction. The critical position to mark the onset of Görtler instability is obtained as a function of the Görtler number, where disturbances at the critical state are mainly confined to the hydrodynamic boundary layer. Comparing the theoretical predictions with available experimental and other theoretical results, the present predictions follow experimental trends fairly well with slightly higher critical Görtler numbers than those from the local stability theory. The propagation theory commanding the local eigenvalue analysis is successful to obtain stability conditions reasonably in Görtler vortex problems, relaxing the limitations by the conventional analyses. © 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction Centrifugal instability in boundary layer flow over concavely curved walls was first demonstrated by Görtler. The instabilities reveal the pattern of counter-rotating streamwise vortices, named as Görtler vortices. Görtler vortices occur when the Görtler number exceeds a critical value [1]. The conventional Görtler number Gθ indicates the ratio of centrifugal to viscous forces by the definition of Gθ =

U∞ θ

ν

r

θ R

,

(1)

where U∞ is the free stream velocity, θ the momentum thickness, ν the kinematic viscosity, and R the concave surface radius of curvature. The basic mechanism on the onset of this vortex is identical to those of Rayleigh and Taylor instabilities in rotating inviscid fluid and rotating viscous one, respectively. The instability motion of Görtler vortices is driven by centrifugal forces associated with change in the direction of motion forced on fluid by the geometry of the boundary. This kind of secondary flow occurs in the wide area of engineering systems, such as high-efficiency curved micromixers [2], parallel plate heat exchangers [3], turbine blades and modern supercritical airfoils [4].

∗

Corresponding author. Tel.: +82 64 754 3685; fax: +82 64 755 3670. E-mail address: [email protected] (M.C. Kim).

0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.07.005

To analyze the onset of secondary motion over concavely curved walls, the local stability analysis has been considered while neglecting the streamwise growth of local disturbances [5]. However, Hämmerlin [6] found that governing equations originated by Görtler cannot reproduce his stability criteria. After Hämmerlin’s report, many theoretical models have been proposed. In determining the critical Görtler number, Floryan and Saric [7] reformulated the same problem by employing the effect of the vertical component of the primary flow. Hall [8] reported that all local stability solutions using the method of separation-of-variables are inadequate for stability conditions in the Görtler problem. He introduced the rate of streamwise growth of disturbances in Görtler vortices. It is plausible to consider the rate of streamwise growth of disturbances in tracking the growth of disturbances at an arbitrary position [9]. Numerous theoretical and experimental results have been reviewed by Herbert [10], Floryan and Saric [11], Floryan [12], Saric [13] and Schlichting and Gersten [14]. The purpose of this study is to determine the onset condition of convective instability in the laminar boundary layer over slightly curved walls by employing the propagation theory, which has been used with success in the transient stability problems of Bénard-type convections [15–17] and in the onset of Taylor-like vortices for unsteady Couette flows [18,19]. The resulting stability conditions based on the local eigenvalue analysis are compared with the available experimental and theoretical results by others.

408

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

following disturbance equations are derived under the assumption of the slightly curved wall (∆/R  1/ReL ) [12]:

∂ U1 ∂ V1 ∂ W1 + + = 0, ∂X ∂Y ∂Z ∂ U1 ∂ U0 ∂ U1 ∂ U0 ∂ U1 + U0 + U1 + W0 + W1 ∂t ∂X ∂X ∂Z ∂Z 1 ∂ P1 =− + ν∇ 2 U1 , ρ ∂X ∂ V1 ∂ V1 ∂ V1 1 ∂ P1 + U0 + W0 =− + ν∇ 2 V1 , ∂t ∂X ∂Z ρ ∂Y ∂ W1 ∂ U0 ∂ W1 ∂ W0 ∂ W1 + U0 + W1 + W0 + W1 ∂t ∂X ∂Z ∂Z ∂Z 1 1 ∂ P1 + ν∇ 2 W1 − 2 U0 U1 . =− ρ ∂Z R

(6)

(7) (8)

(9)

Since the boundary layer thickness is a proper length scale in the boundary layer flow, nondimensionalized variables are introduced as follows:

τ = tU∞ /L,

Fig. 1. Schematic diagram of system considered here.

1/2

2. Theoretical analysis

The system considered here is the laminar boundary layer flow over a slightly curved wall as shown in Fig. 1. The radius of curvature, R is assumed to be much larger than the boundary layer thickness. A Newtonian fluid flows along the X -direction with a free stream velocity U∞ . With the assumption of a slightly curved wall (∆/R  1/ReL ), here ∆ is the boundary layer thickness at a streamwise position L, ReL is the Reynolds number based on the streamwise distance, and the boundary layer assumption of ReL → ∞. Thus, the base flow field can be approximated by the well-known Blasius equation: 1 00 ff = 0, 2 with the following boundary conditions, f 000 +

at ζ = 0,

(2)

p = p /(ρ

(3a)

at ζ → ∞,

∂Ψ ∂Z

and W0 = −

1/2

(3b)

∂Ψ , ∂X

(4)

where U0 and W0 represent base velocities in streamwise and vertical directions, respectively. The justification of the above simplification for the slightly curved wall is illustrated in the experimental results of Liepman [1].

(10)

)/U∞ ,

),

1/4 ReL

√

where G∆ = (L/R) is the Görtler number based on the boundary layer thickness at a given streamwise distance L, i.e. −1/2 LReL . For the present slightly curved wall, G∆ has a small value. It is noted that the streamwise velocity disturbance has a scale 2

of U∞ /G∆ rather than U∞ , since the streamwise disturbances usually have much larger magnitude than spanwise and vertical ones [8,12]. Using the above scales, the dimensionless disturbance equations can be re-written as 1 ∂u G∆ ∂ x 2

where the prime denotes the differentiation with respect to ζ . The √ dimensionless variables are defined by f = Ψ / (ν U∞ X ) and √ ζ = Z (U∞ ν)/X . Here, the stream function Ψ is defined as U0 =

1/2 U0 W0 ReL 1 2 U∞ Re− L

(U , W ) = ( , 0

f0 = 1

1/2

2

(u, v, w) = (G∆ U1 , V1 ReL , W1 ReL )/U∞ ,

2.1. Primary flow

f = f0 = 0

1/2

(x, y, z ) = (X , YReL , ZReL )/L,

+

∂v ∂w + = 0, ∂y ∂z

(11)

∂u ∂u ∂U ∂u ¯ 2 ∂U ∂ 2u ∂ 2u +U +u +W + G∆ w = 2 + 2, ∂τ ∂x ∂x ∂z ∂z ∂y ∂z ∂v ∂v ∂v ∂ p ∂ 2v ∂ 2v +U +W =− + 2 + 2, ∂τ ∂x ∂z ∂y ∂y ∂z ∂w ∂w ∂W ∂w ∂W +U +u +W +w ∂τ ∂x ∂x ∂z ∂z 2 2 ∂p ∂ w ∂ w =− + 2 + 2 − 2Uu, ∂z ∂y ∂z

(12)

(13)

(14)

with the boundary conditions, u=v=w=0

at z = 0 and z → ∞. ∂p ∂2u ∂2v , , ∂ x ∂ x2 ∂ x2

(15) ∂2w ∂ x2

2.2. Disturbance equations

and have been omitted The terms involving 1/ReL , by invoking the boundary layer assumption (ReL → ∞) [12], even

Under the linear stability theory, physical quantities of velocity and pressure are expressed as the linear sum of basic quantities described in the previous section and their infinitesimally perturbed quantities:

though |U1 |  |W1 |, |G∆ ∂ u/∂ x| has the same order of magnitude as |∂w/∂ z | for the region considered here. Since the coefficients of the time-derivative terms are indepen∂u dent of all space coordinates, the time-derivative term ∂τ can be

U = U0 + U1

and P = P0 + P1 ,

(5)

where U = (U , V , W ) denotes the velocity vector in the Cartesian coordinates, and P the pressure. The subscripts 0 and 1 represent the basic and perturbed quantities, respectively. Now, the

−2

∂u represented as ∂τ = σ u under the normal mode analysis, where σ denotes the temporal growth rate. From the theoretical and experimental evidences [20,21], it is known that the primary instability 1/2 is stationary for the region of ReL < O(250), which corresponds

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

to Tollmien–Schlichting instabilities. Therefore the imaginary part of σ , i.e. σi , can be set to zero. As these simplifications may not consider Tollmien–Schlichting instabilities, the amplitude of the fastest growing instabilities are to be found by setting the imaginary part of σ , σr = 0 under the neutral stability condition. Therefore, all terms with regard to time-derivatives are omitted. The primary flow is represented by U = f 0 and W = −x−1/2 { 21 f − 1 2

ζ f 0 }.

for streamwise growth are defined as quantities of streamwise components: r0 =

The propagation theory in finding the critical streamwise position xc to mark the onset of convection postulates that disturbances are propagated mainly within the velocity boundary layer thickness at xc  δ . Here δ is the dimensionless velocity boundary layer −1/2 thickness, δ = ∆/L ∼ x1/2 ReL . In this case, more simplified relations between u and w are obtained from Eqs. (12) and (14) as u ∂U ∼ 2, ∂z δ w u ∼ 2, δ

(16) (17)

a v ∼ w, ∗

1 dE1 2E1 dx

Z

,

(19b)

∞

U 2 dz

E0 =

(18)

where a represents the dimensionless wavenumber based on the boundary layer thickness, i.e. a∗ = aδ . Here a is the spanwise wavenumber based on the unspecified streamwise distance L. From relation (18), it is expected that since w → 0 for the case of a∗ → 0, the system is stable for the long-wave disturbances. The relations of a∗ = aδ, |w/u| = δ 2 and |v/u| = δ/a based on Eqs. (17) and (18) make it possible to construct the transformation functions as u = δ n u∗ , v = δ n+1 v ∗ /a and w = δ n+2 w ∗ . In order to determine the exponent ‘n’ in this stability analysis, a new stability criterion is required. Shen [22] suggested that the growth rate of the perturbation kinetic energy (r1 ) should exceed that of the base flow (r0 ). His suggestion is called the momentary stability condition. According to his concept, the neutral condition in the present system is determined at the position where r0 = r1 . Since the condition of |w/u| → 0 is valid for δ → 0 [9], the dimensionless rates

∞

Z

u2 dz ,

and E1 =

(20)

0

0

where the assumptions of U0  W0 and U1  V1 , W1 are applied. Hall [8] also introduced r1 as a parameter to determine the marginal stability curve. For the case of δ → 0, the stretched vertical coordinate ζ = Z /∆ is more suitable to describe the boundary layer flow [14]. Therefore, we set u(x, z ) = δ n u∗ (x, ζ ). Since the relative stability condition of r0 = r1 is sufficient for n = 0, as explained in the Appendix, we can define the amplitude functions of disturbances as following:

√

where GL = ReL (L/R) is the Görtler number based on the specified streamwise distance L. The above equations are the force balances between the inertia force and the viscous force in the streamwise direction and between the centrifugal and the viscous force in the vertical direction, respectively. These equations mean that the motion of Görtler vortex associated with v and w occurs due to u, and this incipient secondary flow is very weak at x = xc since w ∼ δ 2 u. Because ∂ U /∂ z has the magnitude of 1/δ , it is ¯ 2L δ 3 is a constant for δ  1 from the above known that G∗ = G relations (16) and (17). In this relation the streamwise primary velocity and its perturbation have been nondimensionalized with different scales. If we use the same scaling factor for the streamwise primary velocity and its perturbation, w ∼ u/δ can be obtained under the condition of constant G∗ . Since the motion of Görtler vortex is driven by the centrifugal force related with u, |U1 |  |W1 | has the physical meaning rather than |U1 | ≈ |W1 |. Based on this relation, we choose w ∼ δ 2 u rather than w ∼ u/δ . To examine the stability characteristics of this problem, let us find the minimum value of GL that satisfies Eqs. (11)–(15) for a given x. Now, we assume that at the neutral stability state, the steady disturbances under the principle of the exchange of stabilities [20] are periodical in the spanwise y-direction, having stationary wavenumber from the experimental observations [21,27,28]. From the continuity equation (14), the following relation is obtained: ∗

(19a)

where E0 and E1 are the dimensionless energies of base and disturbance flows, respectively defined by

2.3. Propagation theory

2

1 dE0 2E0 dx

and r1 =

GL w

409



  u

u∗ (ζ )



v  δν (ζ )/a w  = δ 2 w ∗ (ζ )  exp(iay). p δ p∗ (ζ ) ∗

(21)

This means that the amplitude function of streamwise velocity dis∗ turbances follows the behavior of U. Even though we set ∂∂ux = 0,

x-derivative term of ∂ u/∂ x still exists in

∂u ∂x z



=

ζ ζ ∂ u∗ − 2x ∂ζ . Simix

lar relations hold for ∂v/∂ x and ∂w/∂ x. Substituting Eq. (21) into Eqs. (11)–(15), and eliminating ν ∗ and p∗ , we obtain the following set of stability equations: 1

2

(D2 − a∗ )u∗G = − ζ UDu∗G 2 1

− ζ UDu∗G + W ∗ Du∗G + w∗ DU , 2

2

2

3

1

2

2

(22)

(D2 − a∗ )2 w∗ = −2a∗ G∗ Uu∗G − D3 u∗G + ζ D4 u∗G 1 2 1 2 + a∗ Du∗G + ζ a∗ D2 u∗G 2 2   1 1 + DU Dw ∗ − ζ Du∗G 2 2

  1 1 1 − ζ DU D2 w∗ − Du∗G − ζ D2 u∗G 2 2 2   1 1 − ζ U D3 w ∗ − D2 u∗G − ζ D3 u∗G 2 2   1 1 2 ∗ ∗ 2 ∗ ∗ + DW D w − DuG − ζ D uG 2 2   1 + W ∗ D3 w∗ − D2 u∗G − ζ D3 u∗G 2   1 2 2 − U a∗ w ∗ − ζ a∗ D w ∗ 2

2

+ a∗ u∗G



1 2

W∗ +

1 2

 2 2 ζ DW ∗ − W ∗ a∗ Dw∗ − a∗ w∗ DW ∗ ,(23)

with u∗G = w ∗ = Dw ∗ = 0

at ζ = 0 and ∞,

(24)

410

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414 2 GL 3

where u∗G = u∗ /G∗ , a∗ = aδ , G∗ = δ and W ∗ = −{ 21 f − 12 ζ f 0 }. The stability parameters, G∗ and a∗ having dimensionless length scales δ , were already introduced by Hall [8]. The superscript ‘*’ marks the variables defined in (x, ζ )-domain. The above equations can be directly obtained by substituting Eq. (21) into equation (2.9) in Hall’s work [8]. At the neutral state, the base flow dominant with G∗ = constant is still dominant. The parameters a∗ and G∗ scaled by the thickness of the boundary layer are assumed constant under the local stability theory. Even though the stability equation is a sixth order differential equation, an appropriate transformation makes it an eigenvalue problem. Therefore, the present problem might be considered as a generalized eigenvalue problem. From these postulations, the stability equations (22) and (23) with the boundary conditions (24) are solved for the set of eigenvalues for a∗ and G∗ , where u∗G and w ∗ are the corresponding eigenfunctions. In addition, the critical stability condition is determined at the minimum value of G∗ with corresponding a∗ . This procedure is the key concept of the propagation theory in the present stability analysis of the Görtler vortex problem. The conventional local stability analysis under the parallel-flow model omits the terms involving ∂(·)/∂ x and assumes W ∗ = 0 in Eqs. (11)–(14) in amplitude coordinates x and z. This results 2

2

2

in (D2 − a∗ )u∗ = w ∗ DU and (D2 − a∗ )2 w ∗ = −2a∗ G∗ Uu∗ instead of Eqs. (22) and (23) [23]. As discussed above, the crucial difference between the present analysis and the previous ones is

ζ u∗ in the relation of ∂∂ ux z = − 2 ∂∂ζ (see Appendix). Even though x the terms involving ∂ u/∂ x, ∂ν/∂ x and ∂w/∂ x are omitted in most of the previous studies [5–7,10–12,23] under the local stability assumption (r1 = 0), the present model considers these terms as shown in Eq. (21) (r1 = r0 ). It appears that these terms make the system more stable. 2.4. Solution method

To find eigenvalues and eigenfunctions for ordinary differential equations, several methods such as the compound matrix method and the shooting method are proposed [24]. In the present study, for a fixed a∗ , the stability equations (22)–(24) are solved by employing the shooting method. In order to solve the stability equations (22)–(24), the basic flow field solution must be obtained, a priori. For this purpose, the fourth or fifth order Runge–Kutta–Fehlberg method is employed. Now, the proper values of Du∗ , D2 w ∗ and D3 w ∗ at ζ = 0 are assumed for a given a∗ . Since the stability equations and the boundary conditions are all homogeneous, D2 w ∗ at ζ = 0 can be assigned arbitrarily and the parameter G∗ is assumed. This procedure can be understood easily by taking into account the properties of eigenvalue problems. Since all the initial conditions are provided, this initial value problem can proceed numerically. Integration is performed from the plate ζ = 0 to an imaginary outer boundary with the fourth order Runge–Kutta–Gill method. If the guessed values of G∗ , Du∗ (0) and D3 w ∗ (0) are correct, u∗ , w ∗ and Dw ∗ will vanish at this outer boundary. To improve the initial guess the Newton–Raphson iteration is used. When convergence is achieved, the outer boundary is increased by a predetermined value and the above procedure is repeated. Since the disturbances decay exponentially outside the boundary layer, the incremental change in G∗ also decays faster with an increase in outer boundary depth. This behavior enables us to extrapolate the eigenvalue G∗ to the infinite depth. Using a similar procedure, the results from the local stability analysis are obtained. 3. Results and discussion The values of eigenvalues obtained by the above numerical scheme constitute the stability curve, as shown in Fig. 2. The stability criteria of the minimum G∗ are found to be 10.32 with

Fig. 2. Neutral stability curve. The minimum G∗ and corresponding a∗ determine the critical conditions.

its corresponding a∗ value of 0.48. The eigenvalues G∗ and a∗ have the following forms: G∗ = G2 =



∆



R

and a∗ =

Re2∆

√

2π ∆

λ

,

(25)

where G = (Re∆ ∆/R) is the Görtler number based on the boundary layer thickness. Also, Re∆ = (U∞ ∆/ν) is the Reynolds number based on the boundary layer thickness ∆ at some streamwise distance X . λ is the wavelength of the vortex. For the 2 limiting case of a∗ → 0, G∗ a∗ should be constant to keep the

2 stability parameters G∗ and a∗ valid. The resulting asymptotic

−2

relation of G∗ ∼ a∗ for a∗ → 0 is obtained as compared in Fig. 2. This asymptotic relation was also suggested by Hall [8]. Since ∆ has a finite value for the actual boundary layer, the case of a∗ → 0 corresponds to the long-wave disturbance with a large wavelength as seen in Fig. 2. With the relation of θ /∆ = 0.664 [14] and λ/∆ = (2π /0.48) (see Eq. (25)), the critical conditions of G∗ = 10.32 and a∗ = 0.48 yield

  1/2 θ 3/2 θ Gθ = G = Reθ ∆ R √ = 10.32 × 0.6643/2 = 1.738, 

(26a)

and

 λ 3/2 R ∆  3/2 √ 2π = 10.32 × = 152.14, (26b) 0.48 √ where Gθ = (Reθ θ /R) is the Görtler number based on the momentum boundary thickness. Also, Reθ = (U∞ ∆/ν) and Reλ = (U∞ λ/ν) are the Reynolds numbers based on the momentum Λ = Reλ

 1/2 λ



=G

boundary thickness and the wavelength, respectively. Based on the local stability analysis, Görtler [5], Hämmerlin [6] and Herbert [10] analyzed the same problem where they omitted the streamwise growth of local disturbances. Floryan and Saric [11] determined the critical Görtler number by employing the effect of the vertical component of the primary flow from reformulated stability equations by considering the rate of streamwise growth of disturbances as a constant value. Hall [8], and Bottaro and Luchini [25] pointed out that the rate of streamwise growth of disturbances is not constant. They abandoned the conventional normal mode analysis for this problem. Later, Lin and Hwang [26]

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

411

Table 1 Comparison of stability parameters of Gθ and Λ with previous ones.

Present study Lin and Hwang [26] Görtler [22] Hämmerlin [6] Herbert[10] Floryan and Saric [11]

Gθ

Λ

1.738 2.184 0.3138 0.1683 0.3000 0.2509

152.14 160.69 175

∞ ∞ ∞

Fig. 4. Comparison of critical conditions with Bippes’ [27] experimental data. Bippes introduced spanwise periodic disturbance intentionally.

Fig. 3. Comparison of stability condition Gθ with previous experimental and theoretical results. Liepmann’s [1] experimental results are for the large radius of curvature (R = 20 ft), the zero pressure gradient and the low turbulence level (0.06%).

solved numerically Eqs. (11)–(15) under the steady state assumptions. They added arbitrary disturbances having the uniform amplitude of u = 10−3 to the mainstream velocity at x = 0 and solved the linearized Navier–Stokes equation in the (x, ζ )-domain rather than the (x, z )-domain. In Table 1 the critical conditions are summarized. The numerical result is quite close to the present theoretical one, but Gθ -values from other models are much lower than the present one. Liepmann [1] investigated the transition of the boundary layer flow from the laminar to the turbulent region experimentally. He reported the effect of curvature, pressure gradient and free stream turbulence on the transition position. His experimental results for the large radius of curvature (R = 20 ft), the zero pressure gradient and the low turbulence level (0.06%) are compared with the present predictions in Fig. 3. As shown in this figure, theoretical results are lower than experimental data. However, the present prediction by Eq. (26) bounds experimental data more closely than that of [11]. Since linear theory deals only with the growth of infinitesimal disturbances, it is possible that the difference between theory and experiment is due to the fact that only finite disturbances are actually observed. The infinitesimal disturbances must grow appreciably before they are observed. Bippes [27] introduced spanwise periodic disturbances by heating longitudinally oriented wires placed on the surface of the plate with a specific spanwise periodicity. The instabilities were detected by observing changes in the spanwise distribution of hydrogen bubbles. According to the experimental conditions (U∞ , λ and R), the introduced disturbances are excited (amplified) or damped at a certain position. Therefore, the boundary of the excited and damped region will constitute the marginal stability map. His experimental results are compared with the present neutrality curve in Fig. 4. The difference between the present prediction and experimental data may be caused by the free stream turbulence level. According to Liepmann’s [1] experimental results

∗ Fig. 5. Distributions of normalized amplitudes, here |wmax /u∗G,max | = 0.1316. Streamwise velocity disturbances are confined mainly within the boundary layer thickness.

for the smaller radius of curvature (R = 2.5 ft), the critical Gθ value decreases from 9.0 for the lower free stream turbulence level (0.06%) to 6.0 for the higher free stream turbulence level (0.3%). Therefore, it is assumed that our stability curve would be lowered by a certain factor to consider the effect of free stream turbulence. At the critical conditions illustrated above, the amplitude functions of u∗ and w ∗ are featured in Fig. 5, wherein the quantities have been normalized by the corresponding maximum ∗ magnitudes u∗max and wmax . It is seen that streamwise velocity disturbances are confined mainly within the boundary layer thickness, but vertical velocity disturbances are driven more upward over the boundary layer thickness. Experimental uvalues by Swearingen and Blackwelder [21] are compared with the present predictions in Fig. 6. As shown in this figure, our distribution of disturbances represents the experimental results quite well, especially for the region near the curved wall. Based on the distribution of streamwise velocity disturbances, the streamwise growth rate can be obtained from Eq. (19): r0 = r1 = 1/(4xc ). This means that for large ReL , the growth rate at x = xc is inversely proportional to xc . As explained in the Appendix, we choose n = 0 based on the momentary stability concept, r0 = r1 by Shen [22]. If we employ other stability criteria, the value of exponent n can be changed. The effect of exponent n on the neutral stability curve is summarized

412

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

Fig. 6. Comparison of predicted velocity profiles with experimental ones. For the region near the curved wall, the present disturbance represents the experimental one quite well.

Fig. 7. The effect of exponent n on the neutral stability curve. Neutral stability curve can be closer or farther from the experimental data by adjusting n.

in Fig. 7. As shown in this figure, the neutral curve can be closer to the experimental data by adjusting n. However, we determine the value of exponent n considering the physical meaning of the momentary stability concept. Furthermore, it is stressed that the negative value of n does not have a physical meaning, since the quantities of disturbances become infinite as δ → 0. Therefore the present transformation in Eq. (21) seems to bring the most unstable mode physically for the critical G∗ at a minimum value in the neutral stability curve. Hall [8] tracked numerically the streamwise growth of disturbances by exciting disturbances at a specific position xe . He involved the terms of ∂ u/∂ x, ∂ν/∂ x and ∂w/∂ x in the governing equations and solved the linearized partial differential equations in (x, ζ )-domain. He defined the condition of the marginal position xm upon r1 = 0. He introduced the initial disturbances in the form of u = ζ 6 exp(−ζ 2 ) and w = 0 at x = xe

(27)

on the flow. He examined the stability characteristics by tracing whether disturbances grow or not. Even though the local stability has been resolved already, his method struggles to determine the proper initial condition u at the excitation position xe . Some of his stability curves are compared with the present stability curve in

Fig. 8. Comparison of present stability curve with Hall’s [8] simulations. He employed r1 = 0 at x = xc as the stability criterion.

Fig. 8. As shown in this figure, his critical conditions depend on the excitation position xe , and the critical G∗ -value increases with decreasing xe for the case of xe ≤ 20. This may partially explain the difference between his and the present stability condition. Another reason involves in the stability criterion. We employ a relative stability criterion, i.e. r0 = r1 = 1/(4xc ) at x = xc while Hall used r1 = 0 at x = xm as the stability criterion. Since xc > xm , the present critical G∗ -value is larger than his. For the case of xe > 40, his marginal stability curves, except the right-hand branches, are merged. Fig. 9 explains the streamwise growth of disturbances conceptually. This conceptual drawing is based on the numerical simulation results given in Fig. 4 of [8] and Fig. 7 of [9]. They reported that the growth rate during the initial stage is influenced by the excitation position and the introduced initial condition, causing the neutral position to be affected by these factors. For the present system, the fastest growing mode of instability is damped down in the domain of r1 < 0, but it is amplified superexponentially for xc ≤ x ≤ xo , conceptually. At x = xo , r1 reaches the maximum and the secondary flow is detected. Linear theory is applicable to the domain of x ≤ xo . The above scenario may represent the actual phenomena to a certain degree since the initiated disturbances usually have extremely small magnitude. However, the nonlinear effects become important for x ≥ xo . Therefore, the present results are considered as the proper initial conditions for the nonlinear matching problem in a numerical method. The wavelength of Görtler vortices were determined experimentally by Tani and Sakagami [28] and Bippes [27] for the artificially introduced disturbances and by Swearingen and Blackwelder [21] for the naturally introduced disturbances. Their results for the marginal cases are compared with the present results in Fig. 10. As shown in this figure, the present stability curve bounds the experimental data except for lowest four points. As explained previously, the points below the marginal stability curve may be possible by considering the free stream turbulence effect. Since, in Tani and Sakagami’s experiments [28], the free stream turbulence level is of the order of 0.05%, their critical Gθ may be higher than that in the case of the smaller free stream turbulence level. Tani and Sakagami [28] introduced artificial disturbances with a fixed wavelength at a fixed position and determined whether they are excited (amplified) or damped in the streamwise direction. According to their experimental results, disturbances with a certain wavelength are amplified. The corresponding wavelength is shown in Fig. 10. In Bippes’ experiments, the longitudinal vortices are introduced at the leading edge and their first excited positions are

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

413

4. Conclusions The onset of convective instability of the laminar boundary layer flow over a slightly curved wall was analyzed under the non-parallelism of a base flow. New stability equations for disturbances experiencing streamwise variation were derived by employing the propagation theory under the linear stability analysis. The governing parameter was found to be either (∆/R)1/2 Re∆ or (θ /R)1/2 Reθ . Although the present model underpredicts the critical value of (θ /R)1/2 Reθ , the propagation theory explains experimental data fairly well by considering the streamwise variations of disturbances reasonably. It is interesting that the velocity disturbances in the streamwise direction are confined mainly within the boundary layer thickness and they agree well with the experimentally determined disturbance profiles. The present model expects deterministically the Görtler vortex on curved walls is the dominant mode of the instability as the order of G∗ ∼ O(10) competes with the well-known Tollmien–Schlichting wave instability of the 1/2 order of ReL ∼ O(250). The present results may be considered as

Fig. 9. Conceptual diagram of growth of disturbance energy.

1/2

initial conditions in the range from G∗ ∼ O(10) to ReL ∼ O(250) to trace the time-dependent and nonlinear variation of the disturbances in initially steady and linear Görtler vortices. As there are many other factors affecting the onset condition, such as suction, blowing and heating, the present results promise to further resolve the system with similar blowing or suction under the propagation theory. Appendix

√

With the relation of ζ = z / x, the dimensionless energy of base flow defined in Eq. (20) can be reduced as ∞

Z

U 2 dz =

E0 =

√

∞

Z

U 2 dζ .

x

0

(A.1)

0

From the base velocity distribution by Eqs. (2)–(4), U is a function of ζ only, and then r0 is obtained as Fig. 10. Comparison of wavelength parameter Λ with previous experimental results.

measured downstream. Therefore, their experimental points correspond to those of the basic mode in the critical state. Now, the following assumption is introduced: the present critical wavelength is for a basic mode, and a secondary transition in the supercritical flow exists downstream. Over a short distance, alternative pairs of counter-rotating vortices are assumed to grow at the expense of the intervening singles or pairs of vortices according to the way schematized in Fig. 11. Under this assumption, the possible wavelength parameter Λm is obtained from Eq. (26b):

Λm = Remλc



1/2

mλc R

r0 =

.

(A.2)



o , r1 can be reduced as x ( Z  "Z

=

n δ n nδ −1 ddxδ u∗ ζ +



∂ u∗ ∂ x ζ

ζ ∂ u∗ 2x ∂ζ

−

∞

−1

∗2

∞

u dζ

r1 =

u

0

0

∗

nδ

−1

dδ ∗ u dx ζ



) # ∂ u∗ ζ ∂ u∗ + − dζ ∂ x ζ 2x ∂ζ x # Z ∞ −1 "Z ∞ ∗ ∗2 ∗ ∂u = u dζ u dζ ∂ x ζ 0 0

(28)

which is compared with the experimental results in Fig. 10. It is supposed that the case of m = 1 represents the basic mode and those of m = 2 and m = 3 the secondary transition of the basic mode through mechanisms (a) and (b) in Fig. 11, respectively. Similarly, the cases of m = 22 and m = 3 × 2 represent the corresponding secondary transition of m = 2 mode. In Fig. 10, our predictions show that the above illustrations represent naturally occurring and also artificially introduced disturbances. This means that disturbances corresponding to longitudinal vortices having the wavelength given by Eq. (28) are excited over the marginal stability curve, as expected.

4x

And, with the relation of ∂∂ ux z

= 152.14 × m1.5 ,

m = 1, 2, 3, 22 , 3 × 2, . . .

1

+ nδ − 1

dδ dx

+

1 4x

.

(A.3)

From Eqs. (19) and (20), the relative stability condition of r0 = r1 is fulfilled when n = 0 and ∞

Z

∗2

u dζ 0

−1 "Z

(

∞

u 0

∗

) # ∂ u∗ dζ = 0 , ∂ x ζ

(A.4)

or

∂ u∗ = 0, ∂ x ζ

(A.5)

414

M.C. Kim et al. / European Journal of Mechanics B/Fluids 29 (2010) 407–414

a

b

Fig. 11. Schematic diagram of possible mechanisms for second transition: (a) doubled wavelength and (b) tripled wavelength.

because ddxδ 6= 0. This means that the relative stability condition corresponds to the neutral one in the boundary layer (x, ζ )coordinate rather than in the global (x, z )-one. This is the crucial difference between the present model and the previous ones where ∂∂ ux z = 0 is assumed [5–7,10–12]. References [1] H.W. Liepmann, Investigation of boundary layer transition on concave walls, NACA Wartime Report, 4J28, (1945). [2] Y. Yamaguchi, F. Takagi, K. Yamashita, H. Nakamura, H. Maeda, K. Sotowa, K. Kusakabe, Y. Yamasaki, S. Morooka, 3-D simulation and visualization of laminar flow in a microchannel with hair-pin curves, AIChE J. 50 (2004) 1530–1535. [3] R. Toe, A. Ajakh, H. Peerhossaini, Heat transfer enhancement by Görtler instability, Int. J. Heat Fluid Flow 23 (2002) 194–204. [4] S.M. Mangalam, J.R. Dagenhart, T.E. Herpner, J.F. Meyers, The Görtler instability on an airfoil, AIAA paper 85–0491 (1985). [5] H. Görtler, Über den Einfluß der Wandkrümmung auf die Entstehung der Turbulenz, Z. Angew. Math. Mech. 20 (1940) 138–147. [6] G. Hämmerlin, Uber das Eigenwertproblem der dreidimensionalen instabilitat laminar Grenzchichten an konkaven Waden, J. Rat. Mech. Anal. 4 (1955) 279–321. [7] J.M. Floryan, W.S. Saric, Effect of suction on the Görtler instability of boundary layers, AIAA J. 21 (1983) 1635–1639. [8] P. Hall, The linear development of Görtler vortices in growing boundary layers, J. Fluid Mech. 130 (1983) 41–58. [9] H.P. Day, W.S. Saric, Comparing local and marching analysis of Görtler instability, AIAA J. 28 (1989) 1010–1015. [10] Th. Herbert, On the stability of boundary layer along a concave wall, Arch. Mech. Stosow. 28 (1976) 1039–1055. [11] J.M. Floryan, W.S. Saric, Stability of Görtler vortices in boundary layers, AIAA J. 20 (1982) 316–324.

[12] J.M. Floryan, On the Görtler instability of boundary layers, Prog. Aerosp. Sci. 28 (1991) 235–271. [13] W.S. Saric, Görtler vortices, Annu. Rev. Fluid Mech. 26 (1994) 379–409. [14] H. Schlichting, K. Gersten, Boundary Layer Theory, 8th ed., Springer, NY, 1999. [15] M.C. Kim, J.S. Hong, C.K. Choi, The analysis of the onset of soret-driven convection in nanoparticles suspension, AIChE J. 52 (2006) 2333–2339. [16] M.C. Kim, C.K. Choi, J.K. Yeo, The onset of Soret-driven convection in binary mixture heated from above, Phys. Fluids 19 (2007) 084103. [17] M.C. Kim, C.K. Choi, The onset of buoyancy-driven convection in fluid layers with temperature-dependent viscosity, Phys. Earth Planet. Inter. 155 (2006) 42–47. [18] M.C. Kim, C.K. Choi, The onset of Taylor–Gortler vortices during impulsive spindown to rest, Chem. Eng. Sci. 61 (2006) 6478–6485. [19] M.C. Kim, C.K. Choi, Onset of Taylor–Görtler instability in accelerated circular Couette flow, Acta Mech. 188 (2007) 79–92. [20] I.K. Herron, A.D. Clark, Instabilities in the Görtler model for wall bounded flows, Appl. Math. Lett. 13 (2000) 105–110. [21] J.D. Swearingen, R.F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of wall, J. Fluid Mech. 182 (1987) 255–290. [22] S.F. Shen, Some considerations on the laminar stability of time-dependent basic flows, J. Aerosol. Sci. 28 (1961) 397–417. [23] H. Görtler, Instabilität laminarer Grenzschichten an konkaven Wänden gegeüber gewissen dreidimensionalen Störungen, Z. Angew. Math. Mech. 21 (1941) 250–252. [24] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag, NY, 1992. [25] A. Bottaro, P. Luchini, Görtler vortices: are they amenable to local eigenvalue analysis?, Eur. J. Mech. B/Fluids 18 (1999) 47–65. [26] M.H. Lin, G.J. Hwang, Numerical prediction of the formation of Görtler vortices on a concave surface with suction and blowing, Int. J. Numer. Mech. Fluids 31 (1999) 1281–1295. [27] H. Bippes, Experimental study of the laminar-turbulent transition of a concave wall in a parallel flow, NASA TM-75243 (1978). [28] I. Tani, J. Sakagami, Boundary-layer instability at subsonic speeds, in: Proc. Int. Council Aero Sci., 3rd congress, Stockholm, 1962, pp. 391–403.