Bypass Transition of a Boundary Layer subjected to ...

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Bypass Transition of a Boundary Layer subjected to Free-Stream Turbulence F. Péneau1 , H.C. Boisson2 , A. Kondjoyan3 , A. Uranga4 1 - CERAM EAI Tech, rue Albert Einstein - BP 085 - 06902 SOPHIA ANTIPOLIS Cedex FRANCE, [email protected] 2 - IMFT, avenue Camille Soula 31100 Toulouse FRANCE, [email protected] 3 - INRA, Theix Clermont Ferrand FRANCE, [email protected] 4 - UVic, Faculty of Engineering, Victoria British Columbia CANADA V8W 3P6, [email protected]

1.

Introduction

The influence of high free-stream turbulence on the dynamics and heat transfer of boundary layer flows is the source of many experimental and numerical works reviewed in Kondjoyan et al. [9]. Bradshaw [2] put a stone with a series of studies with his team, (Simonich et al. [14], Hancock et al. [7], [8], Baskaran et al. [1]). Bradshaw and his group analyzed the influence of free-stream turbulence not only in terms of turbulence level T u but also in terms of the turbulent length scale Lε. A few years later, Dyban et al.. [5], [4] published a very interesting and original work on the influence of free-stream turbulence on the laminar region of the boundary layer. An increase of 56% in the friction coefficient was observed at low Reynolds numbers (< 20 000) with a free-stream turbulence level of 12.5%. These results underscored the importance of the interaction between the boundary layer and the free-stream near the leading edge. Péneau et al [11] have recovered the same results using large eddy simulation and proposed a first modelisation of the phenomena [12] underlining the extreme importance of the bypass transition of the boundary layer in the near stage. In the present paper we focus on the bypass transition of the boundary layer growing under a free-stream turbulence in the range of 1 to 10 % and compare our numerical results to the experimental work of Matsubara and Alfredsson [10]. After a rapid overview of the numerical model, and the physical characteristics of the problem, we then compare our numerical results to the algebraic or transient growth theory following in that the paper of Matsubara and Alfredsson. In the last part of the paper we draw conclusion on our results and those of Matsubara and Alfredsson. 2.

Numerical method and subgrid modelling

The numerical simulations are carried out using the JADIM code. The three dimensional version of this code has been fully described by Calmet et al [3]. The momentum and scalar equations are discretized using a second-order accurate centered scheme on a staggered grid. The resulting terms are integrated in space on finite volumes and the solution is advanced in time by means of a three-step Runge-Kutta time-stepping procedure. The nonlinear terms of each equation are computed explicitly while the diffusive terms are calculated using the semi-implicit CrankNicholson algorithm. To satisfy the incompressibility conditions, a Poisson equation is solved by combining a direct inversion in the (x1 , x2 ) plane (streamwise and lateral direction) with a spectral Fourier method in the x3 direction (spanwise direction). Concerning the subgrid-scale model used in the present study the starting point is the dynamic mixed model DMM proposed by Zang et al [15]. Large Eddy Simulation equations result from the decomposition of the velocity field into a resolved part ui (computed by the code) and 0 0 a subgrid part ui ; ui (~x, t) = ui (~x, t) + ui (~x, t). The overbar denotes a local spatial average of the variable. In our case, this local spatial average is done implicitly by the discretization of the Navier-Stokes equations. Introducing this decomposition into the Navier-Stokes equations, the 1

½

∂ui /∂xi = 0 ¢ ¡ ∂ui /∂t + ∂ui uj /∂xj = −∂P /∂xi + ∂ 2νS ij − τij /∂xj where S ij = 12 (∂ui /∂xj + ∂uj /∂xi ) denotes the resolved strain rate tensor and ν the kinematic viscosity. The Galilean invariant subgrid-scale tensor for the dynamic field introduced in the above equations (following the definitions of Germano) are τij´ = ui uj − ui uj = Lij + Cij + Rij ³ following system of equations is obtained

0

0

0

0

0

0

0

0

where Lij = ui uj − ui uj , Cij = ui uj + uj ui − ui uj + uj ui and Rij = ui uj − ui uj . In the DMM used for this study, the Leonard terms Lij are calculated explicitly while the cross terms Cij and the Reynolds terms Rij are modeled using the classical concepts of subgrid viscosity νt 2¯ ¯ proposed by Smagorinsky τij − 13 (Rkk + Ckk ) δij = ui uj − ui uj − 2νt S ij with νt = C∆ ¯S ¯ where ¯ ¯ ¡ ¢1 C is a parameter that has to be determined dynamically and ¯S ¯ = 2S ij S ij 2 is the local strain rate. The dynamic computation of the parameters C is based on the approach of Germano [6]. For more details, the reader is referred to Péneau et al [13], [11]. 3.

Boundary conditions and characteristics of the numerical simulation

The configuration under study is a flat plate boundary layer evolving spatially. Taking into account the problem of the boundary conditions, we opt for the following dimensions of the computational domain Lx = 85 · δi , Ly = 50 · δi and Lz = 20 · δi where δi is the displacement thickness at the entrance of the domain of a fully developed turbulent boundary layer. The number of points in each directions is Nx = 96, Ny = 96 and Nz = 64. In the streamwise directions, the first eighty meshes are uniform. The last meshes are progressively dilated to dissipate the strong fluctuations before reaching the exit plane. In wall units the mesh sizes are : (∆x+ , ∆z + ) = (38, 24) in the streamwise and spanwise direction respectively, while in the direction normal to the wall the first point is located£ at y + = 0.18. The ¤ Reynolds number at the entrance, based on δi is Reδi = 1620, while Rex ∈ 4, 6.105 ; 5, 4.105 for a skin friction velocity uτ = 0.0458U∞ where U∞ = 1.23m/s is the mean longitudinal velocity of the free-stream. At the wall, no-slip boundary conditions are imposed while at the top of the domain, a slip boundary condition is imposed on the velocity field u1 = U∞ = 1.23m/s; u2 = 0; u3 = 0. At the exit of the 2 ∂2P domain the outflow condition imposed is ∂∂nu2i = 0, ∂x∂y = 0. Concerning the entrance boundary condition which is at the heart of the present numerical simulation, a full presentation of how the entrance signals are generated is presented in Péneau et al [12]. Figure 1 gives you an idea of the method followed to performed the spatially evolving numerical simulation by consecutive calculations.

Figure 1: Method to performed spatial large eddy simulation by consecutive calculations

2

Tu %

Iu %

Case A

1

2

Case B

3

5

Case C0

3

5

Case D

5

8

Case E

10

16

Table 1: Turbulent characteristics of the five free-stream turbulent fields generated 4.

Results

The results presented thereafter are based on large eddy simulation of 5 different cases of freestream turbulence fields hitting the leading edge of a flat plate. Those fields have been fully characterized in Péneau et al [12]. Table 1 present the turbulent characteristics of those fields. The analysis of figures 2 to 6 for case C namely Tu = 3% (figures (a)) show that the large eddy simulations predict very well the experimental results presented by Matsubara et al [10] for a turbulent field of Tu = 1.5%. It seems indeed that even for this higher Tu the boundary layer goes through all its main stages, namely the non-modal growth region, which is shorter than for Matsubara et al but seems to be of equivalent size for case A (see fig. 6), the secondary instability stage, the intermittent spot region and the fully turbulent boundary layer region. Figure 3 shows the deviation from the Blasius form of the velocity profile. and like Matsubara et al it shows a pretty good collapse for different downstream position. This collapse seems to be even better for the case E Tu = 10%. In Figure 4 and 5 we observed the same evolution of the urms profile than Matsubara et al for the case C with the shift of the peak of urms towards the wall, the decrease of the maximum value accompanied by the creation of the logarithmic region revealed through the flatening of the profiles for 1 ≤ δy∗ ≤ 2, where δ ∗ is the theoretical laminar boundary layer displacement thickness. For case E if the shift of the maximum value towards the wall is observed, it seems that the boundary layer wake and logaritmic region is not as well organize than in case C. Moreover, if for both cases C and E, we observe a linear growth of u2rms with the downstream position, the increase is much faster for Tu = 10% and we do not observed the constant position of the maximum value during the linear growth. So we can say that even if a larger Tu does accelerate the transition to turbulence, the fully developed turbulent boundary layer is not especially reached sooner. We can indeed observe in figure 5 that the repartition of the turbulent energy in the boundary layer in case E Tu = 10 % is quite different from case C Tu = 3%. The maximum value of urms is smaller for the larger Tu but urms is higher in the logarithmic and wake region. The analysis of figure 6 underlines the different stage through which the boundary layer passes before becoming fully turbulent. It also reveals the importance of the length scale of the free-stream turbulent field. Indeed comparing case B and C which have the same Tu, we see through the value of the shape factor H that with a larger length scale the free-stream turbulent field has a bigger influence on the boundary layer transition. In figure 7 we present movies showing the evolution of the streaky structures in the boundary layer for the case E. The intermittency of the ejection phenomena can be observed for Reynolds number less than 80 000 and a slight thickening of the streaks can also be observed between the two animations. The BIPOD analysis of the field near the wall, not presented there for a lack of space, confirm this. The characteristic lengths of the streaky structure readed on the BIPOD analysis are similar to those observed by Matsubara et al. 0

Case C was generated with bigger vortices : Lε ca se

C

3

> Lε ca se

B

Figure 2: Mean velocity profile (U); a) Case C Tu ' 3% b) Case E Tu ' 10%

Figure 3: Deviation of the mean velocity profile from Blasius solution normalized with its maximum value; a) Case C Tu ' 3% b) Case E Tu ' 10%

Figure 4: urms distribution; a) Case C Tu ' 3% b) Case E Tu ' 10% 4

Figure 5: ¥ maximum of urms ;H position of maximum of urms as a function of Rex ; a) Case C Tu ' 3% b) Case E Tu ' 10%

Figure 6: Evolution of the shape factor H with the downstream positions from the leading edge

Click on the figure to open animation

Click on the figure to open animation

Figure 7: Flow visualization in time of streaky structures in boundary layers Case E Tu ' 10% 5

References [1] V. Baskaran, O. E. Abdellatif, and P. Bradshaw. Effects of free-stream turbulence on turbulent boundary layers with convective heat transfer. In Stanford University, editor, Seventh Symposium on Turbulent Shear Flows, 21-23 August 1989. [2] P. Bradshaw. Effect of free-stream turbulence on turbulent shear layers. I. C. Aero Report 74-10, Imperial college of science and technology Department of Aeronautics, Prince consort road, London SW7 2BY, October 1974. [3] I. Calmet and J. Magnaudet. Large eddy simulation of high-schmidt number mass transfer in a turbulent channel flow. Phys. Fluids, 9(2):438—455, 1997. [4] E. P. Dyban and E. Ya. Epick. Transferts de Chaleur et Hydrodynamique Dans Les Écoulements Rendus Turbulents. Monagraphie Traduite du Russe à l’I.N.R.A. (traduction N. Zuzine, révision A. Kondjoyan), disponoble au laboratoire, 1985. Monagraphie Traduite du Russe à l’I.N.R.A. (traduction N. Zuzine, révision A. Kondjoyan), disponible au laboratoire. [5] E. P. Dyban, E. Ya. Epik, and T. T. Surpun. Characteristics of the laminar layer with increased turbulence of the outer stream. International chemical engineering, 17(3):501— 504, July 1977. [6] M. Germano. Turbulence: The filtering approach. J. Fluid. Mech., 238:325—336, 1992. [7] P. E. Hancock and P. Bradshaw. The effect of free-stream turbulence on turbulent boundary layers. Journal of Fluids Engineering, 105:284—289, 1983. [8] P. E. Hancock and P. Bradshaw. Turbulence structure of a boundary layer beneath a turbulent free stream. J. Fluid Mech., 205:45—76, 1989. [9] A. Kondjoyan, F. Péneau, and H.C. Boisson. Effect of high free-stream turbulence on heat transfer between plates and air flows : A review of existing experimental results. Int. J. Therm. Sci., 41:1—16, 2002. [10] M. Matsubara and P. H. Alfredsson. Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech., 430:149—168, 2001. [11] F. Péneau, H.C. Boisson, and N. Djilali. Large eddy simulation of the influence of high free-stream turbulence on spatially evolving boundary layer. Int. J. of Heat and Fluid Flow, 21:640—647, 2000. [12] F. Péneau, H.C. Boisson, A. Kondjoyan, and N. Djilali. Structure of a flat plate boundary layer subjected to free-stream turbulence. Int. J. of Computational Fluid Dynamics, 18(2):1—14, February 2004. [13] F. Péneau, D. Legendre, J. Magnaudet, and H. C. Boisson. Large eddy simulation of a spatially growing boundary layer using a dynamic mixed subgrid-scale model. Symposium ERCOFTAC on Direct and Large-Eddy Simulation, Cambridge May 1999, 12-14 may 1999. [14] J. C. Simonich and P. Bradshaw. Effect of free-stream turbulence on heat transfer through a turbulent boundary layer. Transaction of the ASME Journal of Heat Transfer, 100:671—677, NOVEMBER 1978. [15] Y. Zang, R. L. Street, and J. R. Koseff. A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids, A5(12):3186—3196, 1993.

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