Applications of parabolized stability equation for predicting transition position in boundary layers

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1 Appl. Math. Mech. -Engl. Ed., 33(6), (2012) DOI /s c Shanghai University and Springer-Verlag Berlin Heidelberg 2012 Applied Mathematics and Mechanics (English Edition) Applications of parabolized stability equation for predicting transition position in boundary layers Jia LI ( ), Ji-sheng LUO ( ) (Department of Mechanics, Tianjin University, Tianjin , P. R. China) Abstract The phenomenon of laminar-turbulent transition exists universally in nature and various engineering practice. The prediction of transition position is one of crucial theories and practical problems in fluid mechanics due to the different characteristics of laminar flow and turbulent flow. Two types of disturbances are imposed at the entrance, i.e., identical amplitude and wavepacket disturbances, along the spanwise direction in the incompressible boundary layers. The disturbances of identical amplitude are consisted of one two-dimensional (2D) wave and two three-dimensional (3D) waves. The parabolized stability equation (PSE) is used to research the evolution of disturbances and to predict the transition position. The results are compared with those obtained by the numerical simulation. The results show that the PSE method can investigate the evolution of disturbances and predict the transition position. At the same time, the calculation speed is much faster than that of the numerical simulation. Key words transition position, incompressible boundary layer, parabolized stability equation (PSE), numerical simulation Chinese Library Classification O Mathematics Subject Classification 76F25 1 Introduction The prediction of transition position is one of important research subjects in fluid mechanics. Two flow states of laminar flow and turbulent flow exist in boundary layers. The friction resistance, heat diffusion, and growth of boundary layer thickness in laminar flow are smaller than those in turbulent flow. Therefore, the prediction of laminar-turbulent transition is very important in the design of certain flying vehicles. Today, a few methods have been used to predict the transition position. The e N method and transition modes are usually employed. The e N method is based on the linear stability theory. It needs a lot of experience to choose the value of N. A lot of coefficients in transition modes need to be adjusted in accordance with the experiment results. Herbert [1 2], Bertolotti and Herbert [3], and Chang et al. [4] proposed the parabolized stability equation (PSE) method to research the evolution of disturbances. This method takes the nonparallel effects and the nonlinear evolution of disturbances into consideration. It does not need any experience in the prediction of the transition position. Besides, the computational Received Jul. 20, 2011 / Revised Mar. 19, 2012 Project supported by the National Basic Research Program of China (No.2009CB724103) Corresponding author Ji-sheng LUO, Professor, Ph.D., jsluo@tju.edu.cn

2 680 Jia LI and Ji-sheng LUO storage and time can be reduced by using this method [5]. Based on these merits, the PSE method has been widely used to research the evolution of disturbances. There are a lot of works for the applications of the PSE in incompressible flow. Joslin et al. [6] showed that the PSE could exactly predict the evolution of two-dimensional (2D) Tollmien- Schlichting (T-S) waves and the transition of subharmonic instability waves and oblique waves. Esfahanian et al. [7] used the PSE method to do the stability analysis and predict the transition position for the incompressible flat plate. The disturbances imposed at the entrance were consisted of one 2D wave and a pair of three-dimensional (3D) waves. The results showed that the PSE could research the evolution of disturbances and predict the transition position. However, in their research, the disturbances introduced at the inlet were composed of several waves, and these situations were simple relatively. In nature, the forms of disturbances are very complicated. The wavepacket disturbance is one form of disturbances that commonly exists, and it is more complicated than the combination of several simple disturbances. Therefore, it is significant to research the evolution of wavepacket disturbances. The PSE is used to investigate the evolution of disturbances. When the effects of nonlinearity become drastic, showing that the calculation breaks down or the skin-friction coefficient begins to rise, the station is regarded as the start of the transition process. The results are compared with those obtained by the numerical simulation. 2 Computational model and basic flow 2.1 Computational model A plate boundary layer model in the Cartesian coordinates is shown in Fig. 1. The fluid flows along the plate. x is the flow direction, y is the normal direction, and z is the spanwise direction. The corresponding velocity components are u, v, and w separately. As shown in the figure, the origin of the coordinates is fixed at x=x 0, and U is the free-stream velocity from far away. Fig. 1 Flat plate boundary layer model sketch 2.2 Basic flow The basic flow can be obtained by solving the Blasius equation. The formula and the boundary conditions are as follows: f (η) + c2 0 2 f (η) f (η) = 0, f (0) = f (0) = 0, η = 0, (1) f ( ) = 1, η =, where η = y/δ. δ is the local displacement thickness defined as δ = c 0 ν (x 0 + x) U, δ 0 = c 0 νx0 U, (2)

3 Applications of PSE for predicting transition position in boundary layers 681 where ν is the kinematic viscosity and constant c 0 = We define the Reynolds number Re as follows: Re = U δ 0. (3) ν Using the Runge-Kutta scheme, we can resolve f (η), f (η), and f (η). The relations between u, v, and f (η) are shown as follows: u = U f (η), v = c2 0 U δ 0 2Reδ (ηf (η) f(η)). 3 Governing equations and numerical methods In this paper, the disturbance equations can be obtained from the full Navier-Stokes (N-S) equations. Then, the stability equations are derived with the characteristics of disturbances being taken into account. Finally, the equations are parabolized to become the PSE. To make the equations non-dimensional, the free stream velocity U is used as the reference velocity, and δ 0 is used as the reference length scale, where δ 0 is the displacement thickness at the entrance. The instantaneous variables are decomposed as the sum of the steady basic flow and the disturbance as follows: u i = U i + u i, p = P + p, (5) where U i and P denote the basic flow, and u i and p mean the disturbance. Substituting Eq.(5) into the N-S equations and subtracting the equations corresponding to the steady basic flow, the disturbance equations are as follows: u j = 0, x j u i t + U j u i U i u i + u j + u j = p + 1 ( x j x j x j x i Re x j ) u i, i = 1, 2, 3. x j Considering the evolution of the disturbance in space, the disturbance is assumed to have the following form [8] : M N φ(x, y, z, t) = ˆφ mn (x, y)e i(r x α x 0 mn(x)dx+nβ 0z mω 0t) + c c, (7) m= M n= N where ˆφ mn = (û, ˆv, ŵ, ˆp) T denotes the shape function vector, α mn is the streamwise wave number, nβ 0 is the spanwise wave number, mω 0 is the frequency, and c c is the complex conjugate. Both the shape function ˆφ mn and the wave number α mn are slowly varying functions of x. Substituting Eq.(7) into the disturbance equations (6) and neglecting the second derivative of the shape function along the streamwise direction, we can obtain the nonlinear PSE as follows: L 0 (û, ˆv, ŵ, ˆp) T + L 1 x (û, ˆv, ŵ, ˆp)T = f, (8) where f is the nonlinear term. L 0, L 1, and f are as follows: L 0 = ( K 3 + iα K 2 U x V x W x ) K 3 + y U y ( K 2 V y W y ) K 3 + inβ 0 U z V z ( K 2 W z ) iα y inβ, (4) (6)

4 682 Jia LI and Ji-sheng LUO where K L 1 = 0 K 1 0 0, f = 0 0 K u û x + v û y + w û z u ˆv x + v ˆv y + w ˆv z u ŵ x + v ŵ y + w ŵ z K 2 = 1 ( i dα ) Re dx α2 1 Re n2 β0 2 iαu inβ 0 W + imω 0, K 1 = 2iα Re U, K 3 = 1 2 Re y 2 V y. When these equations are solved, for the streamwise derivative, the first-order difference is adopted in the first step, then the second-order in the second step and the third-order difference in the later. The equation is given by L 1 γ 0 xûn+1 + L 0 û n+1 = 2 k=0, ( L 1 1 x σ kû n k + ζ k f n k), (9) where the values of γ 0, σ k, and ζ k are different according to the different precisions. The values are listed in Table 1. Table 1 Coefficients of different orders γ 0 σ 0 σ 1 σ 2 ζ 0 ζ 1 ζ 2 First-order Second-order 3/2 2 1/ Third-order 11/6 3 3/2 1/ In the y-direction, the fourth-order compact finite-difference proposed by Malik et al. [9] is adopted. The no-slip boundary condition is used at the wall such that u = v = w = 0 at y = 0. The boundary condition outside the boundary layer is u = v = w = 0 when y. The variable space grids are used in the y-direction, and the transformation relation has the following form: L y R j y j = ( ), (10) 1 + C b 1 R 2 j where L y is the computation length in the normal direction. R j = (j 1) / (j n 1) denotes the calculation coordinate, and j n is the number of grid in the normal direction. C b is a constant, C b = ( 1 + 8k 2 3 ) /4. k 2 denotes the ratio between two ending grid space values, k 2 = (y jn y jn 1)/(y 2 y 1 ). 4 Calculation results and discussion Two different types of cases are studied in this paper. One is the identical amplitude disturbances along the spanwise direction. The disturbances at the entrance are consisted of one 2D wave and two 3D waves. The cases of different amplitudes of the 2D wave are calculated. The other is the wavepacket distribution along the spanwise direction, that the disturbances are T-S waves modulated by the Gauss distribution. The results of the PSE and the numerical simulation are shown for these two types of disturbances, and the results are compared and analyzed.

5 Applications of PSE for predicting transition position in boundary layers Results of identical amplitude disturbance along spanwise The disturbances of one 2D wave and two 3D waves are introduced at the entrance. According to the different amplitudes of the 2D wave, three cases are calculated. The parameters are selected, where the Reynolds number Re=732, the basic frequency ω 0 = , and the basic spanwise wave number β 0 = The form of disturbances is as follows: A 2 u 2 (y)e i(α2x ω2t) + A 3 (u + 3 (y)ei(α3x+β3z ω3t) + u 3 (y)ei(α3x β3z ω3t) ) + c c, (11) where ω 2 and ω 3 are the frequencies of the 2D and 3D waves, respectively. β 3 is the spanwise wave number to the 3D waves. α 2 and α 3 are the streamwise wave numbers of the 2D and 3D waves. u 2 (y) and u ± 3 (y) are the solutions to the stability equations. A 2 and A 3 are the amplitudes of the disturbances. ω 2 = 2ω 0, ω 3 = 2ω 0, and β 3 = β 0. The results of α 2 =( , ) and α 3 =( , ) are obtained by the stability analysis. A 3 = and A 2 equals 0.01, 0.006, or The results of the three cases by using the PSE method are shown in Fig. 2, where the DNS presents the results of the numerical simulation. The evolution curves of the 2D wave along the streamwise are shown in Fig. 2(a). The evolutions of the skin-friction coefficient C f are shown in Fig.2(b). The amplitude evolutions of the 2D and 3D basic waves are shown in Fig. 2(c). Fig. 2 Results of different wave amplitudes

6 684 Jia LI and Ji-sheng LUO The evolution results of the disturbances obtained by the PSE and the numerical simulation are shown in Fig. 2(a). The PSE can research the evolution of disturbances since the results between the PSE and the numerical simulation are consistent. It is shown in Fig. 2(b) that there is a small discrepancy between two methods for the skin-friction coefficient. If the rise of C f is seen as the transition position, then the transition position is x = 340 for the case of A 2 = 0.01, the transition position is x = 440 for the case of A 2 = 0.006, and the transition position is x=560 for the case of A 2 = Therefore, the evolution of disturbances and the prediction of transition position can be researched by the PSE method with the wavepacket disturbances. In Fig. 2(c), the amplitude of the 2D basic wave reaches its maximum value and then decreases before transition. When the amplitude of the 2D wave decreases, the amplitudes of 3D basic waves increase quickly, and the transition occurs soon. 4.2 Results of wavepacket disturbance along spanwise With the purpose to study more complex disturbances, the wavepacket disturbances are introduced at the entrance in this paper. The cases of different Reynolds numbers and different frequencies are calculated by using the PSE. The results are compared with those obtained by the numerical simulation to verify the PSE procedure. The wavepacket disturbances at the inlet are T-S waves modulated by the Gauss distribution. The form of disturbances is as follows: Lz a(z e 2 )2 (A 2 u 2 (y)e i(α2x ω2t) +A 3 (u + 3 (y)ei(α3x+β3z ω3t) +u 3 (y)ei(α3x β3z ω3t) )+c c ), (12) where A 2 and A 3 are the amplitudes of the 2D and 3D waves, respectively. The eigenfunctions of u 2 and u 3 are corresponding to the eigenvalues α, β, and ω. The selected parameters are a=0.002, A 2 =0.01, and β 0 = Four cases are calculated and the parameters of these cases are given as follows: A : Re = 732, ω 2 = , A 3 = 0; B : Re = 800, ω 2 = , A 3 = 0; C : Re = 932, ω 2 = , A 3 = 0; D : Re = 732, ω 2 = , A 3 = The neutral curve of the 2D wave and the maximum growth curve of disturbances are shown in Fig. 3. The positions of these cases are also shown. From Fig. 3, the growth rates of selected cases are less than the maximum, and it ensures the disturbances growing fully. Based on these results, the transition can be simulated. Fig. 3 Neutral curve (solid) and maximum growth curve (dashed) The evolution curves of the disturbance velocity u and the skin-friction coefficient C f are shown in Fig.4, where the DNS presents the results of the numerical simulation.

7 Applications of PSE for predicting transition position in boundary layers 685 Fig. 4 Disturbance velocity and skin-friction coefficient C f of different cases

8 686 Jia LI and Ji-sheng LUO In the four cases, before the transition, the results of evolution of disturbances used by the PSE coincide with those obtained by the numerical simulation method. This indicates that the evolution of disturbances can be researched by using the PSE. Near the transition position, the nonlinear effect is very powerful and the calculation by using the PSE will diverge when the amplitude evolution of disturbances changes very sharply. In the curves of the skin-friction coefficient, the results of the PSE and the numerical simulation are accordant. Therefore, the PSE method can predict the transition position with the wavepacket disturbances introduced at the inlet. 5 Conclusions In this paper, the research object is a 3D incompressible boundary layer. The 2D laminar Blasius similarity solution is taken as the basic flow. The evolution and transition of disturbances are researched by using the nonlinear PSE and the numerical simulation with identical amplitude and wavepacket disturbances imposed at the entrance. The following conclusions are obtained by comparing the evolution curves of the 2D wave and the skin-friction coefficient calculated by the PSE and numerical simulation methods. (i) In the linear stage of evolution, the results by using the PSE are consistent with those obtained by the numerical simulation. Therefore, the PSE can be used to research the evolution of disturbances. (ii) Combined with the results by using the numerical simulation method, when the nonlinear effect of disturbances is very powerful, the skin-friction coefficient begins to rise, or the calculation breaks down, the station can be seen as the start of the transition process. (iii) The results obtained by the PSE indicate that the amplitude of the 2D wave increases firstly, and then decreases before transition. While the 2D wave decrease, the 3D waves grow sharply, and the transition occurs soon. The transition is directly related to the increase in the 3D waves, and the 2D wave only plays the roles of auxiliary and promotion. References [1] Herbert, T. H. Boundary-layer transition-analysis and prediction revisited. AIAA Paper (1991) [2] Herbert, T. H. Parabolized stability equation. Annual Review of Fluid Mechanics, 29(1), (1997) [3] Bertolotti, F. P. and Herbert, T. H. Analysis of linear stability of compressible boundary layers using the PSE. Theoretical and Computational Fluid Dynamics, 3(2), (1991) [4] Chang, C. L., Malik, M. R., Erlebacher, G., and Hussaini, M. Y. Compressible stability of growing boundary layer using parabolized stability equations. AIAA Paper (1991) [5] Bertolotti, F. P., Herbert, T. H., and Spalart, P. R. Linear and nonlinear stability of the Blasius boundary layer. Journal of Fluid Mechanics, 242(1), (1992) [6] Joslin, R. D., Chang, C. L., and Streett, C. L. Spatial direct numerical simulation of boundarylayer transition mechanisms: validation of PSE theory. Theoretical and Computational Fluid Dynamics, 4(6), (1993) [7] Esfanhanian, V., Hejranfar, K., and Sabetghadam, F. Linear and nonlinear PSE for stability analysis of the Blasius boundary layer using compact scheme. Journal of Fluids Engineering, 123(3), (2001) [8] Zhang, Y. M. and Zhou, H. Verification of the parabolized stability equations for its application to compressible boundary layers. Applied Mathematics and Mechanics (English Edition), 28(8), (2007) DOI /s [9] Malik, M. R., Chuang, S., and Hussaini, M. Y. Accurate numerical solution of compressible, linear stability equation. Journal of Applied Mathematics and Physics (ZAMP), 33(2), (1982)

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