Structure of a Flat Plate Boundary Layer Subjected to Free-Stream Turbulence

The structure near the flat plate leading edge of an incompressible boundary layer subjected to free-stream turbulence is investigated using large eddy simulation (LES) with a dynamic mixed subgrid-scale model. Free-stream turbulent intensities ranging from 1.5 to 10% are investigated. The evolutions of the velocity, temperature and turbulent intensity profiles inside the boundary layer and the characteristic length scales are analyzed and compared to a laminar flat plate boundary layer. The impact of the free-stream turbulence is examined for Reynolds numbers ranging from 0 (leading edge of the flat plate) to by using two consecutive spatial LESs. Bypass transition is found to occur in the early stage of the boundary layer development for all five cases presented here. The analysis indicates that the thermal turbulent structures transfer more energy and hence generate more thermal fluctuations in their wake than the dynamic structures.


C f
Friction coefficient with free-stream turbulence C f 0 Friction coefficient without free-stream turbulence D ¼ l rC p Thermal diffusivity D t Subgrid diffusivity, turbulent diffusivity ¼ kv 0 T 0 l=›T=›y Stanton number without free-stream turbulence T ¼ T mean þ T 0 Reynolds decomposition for the temperature field T mean Mean temperature T 0 Temperature fluctuations kT 0 l ¼ 0 T rms ¼ kT 0 T 0 l 0:5 Temperature turbulent level Large eddy decomposition for the temperature field T Resolved temperature field T 00 Subgrid temperature field

Reynolds decomposition
for the velocity field U i Mean velocity field; U 1 ¼ U longitudinal direction; Mean free-stream longitudinal velocity component Large eddy decomposition for the velocity field u i Resolved velocity field u 00 i Subgrid velocity field u rms ¼ ku 0 1 u 0 1 l 0:5 Longitudinal turbulent fluctuations v rms ¼ ku 0 2 u 0 2 l 0:5 Lateral turbulent fluctuations w rms ¼ ku 0 3 u 0 3 l 0:5 Span-wise turbulent fluctuations d Dynamic boundary layer thickness d T Thermal boundary layer thickness Average operator in the spanwise direction and in time

INTRODUCTION
The influence of high free-stream turbulence on the dynamics and heat transfer of boundary layer flows is relevant to many industrial applications and has been the subject of experimental and theoretical research for some time. The first studies were performed in the thirties by Fage and Falkner (1931) who did not observe any influence on heat transfer in the presumed laminar region. Due to discrepancies and contradictions in reported findings, the field remained wide open until the landmark work of Bradshaw (1974). In a series of studies with his team, (Simonich and Bradshaw, 1978;Bradshaw, 1983, 1989;Baskaran et al., 1989), Bradshaw and his group analyzed the influence of free-stream turbulence not only in terms of the turbulence level Tu but also in terms of the turbulent length scale L e . It was found that if L e is too small with respect to the boundary layer length scale, then the freestream turbulent structures dissipate before reaching the surface of the plate with no increase in the dynamic activities of the flow or in heat transfer. In his report of 1974, Bradshaw noted the importance of the work of Charnay et al. (1971Charnay et al. ( , 1972Charnay et al. ( , 1976 on the interaction mechanisms between the free-stream flow and the wake of the boundary layer. A few years later, Dyban et al. (1977), Dyban and Epick (1985) published a very interesting and original work on the influence of free-stream turbulence on the laminar region of the boundary layer. With the focus of both academic and industrial research at that time on turbulent flow, Dyban et al.'s work did not receive the attention it deserved. 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. Meanwhile, a consensus has emerged from the numerous studies on turbulent boundary layers. Free-stream turbulence does increase the dynamic and heat transfer in the turbulent boundary layer and heat transfer seems to be more sensitive to free-stream turbulence (Pédisius et al., 1979;Meier and Kreplin, 1980;Blair, 1983a,b). Although this conclusion is general, there is considerable variation from study to study on the magnitude of the heat transfer enhancement. Several studies in the 80s and 90s attempted to model these phenomena with varying success (Young et al., 1992;Maciejewski and Moffat, 1992). A review of the main experimental work can be found in Kondjoyan et al. (2002).
In this paper we present a numerical analysis of the influence of free-stream turbulence on the "laminar" boundary layer. Our results are compared to the experimental data of Dyban et al. (1977) and Dyban and Epick (1985) and we propose an interpretation of the observed phenomena. The paper provides first an overview of the numerical model, the computational procedure and the physical characteristics of the problem. We then present the evolution of the velocity and temperature profiles inside the boundary layer in the presence of free-stream turbulence. The turbulent intensity and turbulent energy production profiles are examined and compared to those for a turbulent boundary layer with no free-stream turbulence. Finally, a physical interpretation of the results is proposed based on an analysis of the Reynolds stresses and turbulent Prandtl number.

NUMERICAL METHOD AND SUBGRID MODELING
The LES Equations and the Subgrid-scale Model The LES equations are obtained by introducing the following decomposition in the Navier -Stokes equations: where u i represents the implicitly filtered, resolved velocity field, and u 00 i represents the subgrid velocity field. This yields: with L ij is calculated explicitly, while C ij and R ij are modeled using the classical concept of subgrid viscosity n t proposed by Smagorinsky (1963) in conjunction with the dynamic mixed model (DMM) of Zang et al. (1993): With this model n t is a function of space and time. C kk and R kk are incorporated into the pressure terms because of the incompressibility condition. For further details on the DMM and the calculation procedure for evaluating n t , the reader is referred to Péneau et al. (1999). The introduction of a similar decomposition into the energy equation yields: with Again, Lc j is calculated explicitly while Cc j and Rc j are modeled using The Numerical Method The numerical simulations are carried out using JADIM code (Calmet and Magnaudet, 1996). The momentum and scalar equations are discretized using a finite volume method with a second-order centered scheme on a staggered grid. The solution is advanced in time using a three-step Runge-Kutta procedure. The nonlinear terms are computed explicitly while the diffusive terms are calculated using the semi-implicit Crank-Nicholson algorithm. To satisfy the incompressibility condition, a Poisson equation is solved by combining a direct inversion in the (x,y) (longitudinal and lateral direction, respectively) plane with a spectral Fourier method in the third direction z (transversal direction). For more details on JADIM the reader is referred to Calmet and Magnaudet (1996).

Grid and Physical Parameter
The grid on which the spatial simulations are performed was initially designed to simulate a fully turbulent boundary layer with entrance Reynolds number, based on the displacement thickness d 1 , Re d 1 ¼ 1620: The dimensions of this domain are L x ¼ 85d 1 ; L y ¼ 50d 1 and L z ¼ 20d 1 with the following number of mesh points in the three direction N x ¼ 96; N y ¼ 96 and N z ¼ 64. In wall units, the mesh size in the longitudinal and span-wise directions is Dx þ ¼ ðDx·u t Þ=ðnÞ ¼ 38 and Dz þ ¼ 24: In the previous equation u t is the friction velocity of a fully turbulent boundary layer at Re d 1 ¼ 1620: Near the wall, the mesh is refined in the normal direction, with the first point located at y þ ¼ 0:18: The physical dimensions are L x ¼ 1:7 m; L y ¼ 1 m; L z ¼ 0:4 m and Dx ¼ 1:04 £ 10 22 m and Dz ¼ 0:625 £ 10 22 m. This mesh, which proved to be well adapted for the simulation of a fully turbulent boundary layer (Péneau et al., 1999), is used to simulate the development under an air flow ðPr ¼ 0:72Þ boundary layer near the leading edge of a flat plate subjected to a free-stream turbulent field. The mean free-stream velocity is U e ¼ 1:23 m=s: The free-stream temperature is T e ¼ 273 K while the wall temperature is set to T w ¼ 274 K: Open boundary conditions are set as › 2 u i =›n 2 ¼ 0; where n is the direction normal to the exit plane.

THE HIGH FREE-STREAM TURBULENCE FIELD AND THE SPATIAL SIMULATION OF THE BOUNDARY LAYER
The generation of the turbulent fields used as entrance conditions for our spatial large eddy simulation (LES) has been fully presented in Péneau et al. (2000). The same method was used here, utilizing the computational mesh described in the preceding section so that a larger range of length scale is covered and a better resolution of the vortices is achieved in the region near the wall. The resulting free-stream turbulence field as well as the boundary conditions used for the simulation are shown in Fig. 1. The velocity field corresponding to the middle yz plane is recorded for 1000 time steps, and is then used as an entrance condition for the spatial simulation of the boundary layer development.
Five cases of free-stream turbulence have been generated this way. Table I presents their turbulent characteristics. Figures 2 and 3 show the turbulence level and intensity profiles for the five cases of free-stream turbulent fields investigated, and Fig. 4 shows the temporal evolution. The profiles presented were obtained by applying an averaging operator in the span-wise direction and in time (1000 time steps). "y ¼ 0 m" corresponds to the position of the flat plate while "y ¼ 1 m" corresponds to the north boundary of the domain of calculation. In this direction, the mesh size increases progressively and hence there is a decrease in the turbulent levels and intensities as we reach the north boundary.
This phenomenon is less important for cases D and E because these turbulent fields were generated with larger vortices which are better resolved and do not dissipate as much in this region. The root mean square velocity profiles for the five cases indicate that the turbulent fields are almost isotropic and homogeneous except near the north and south boundaries of the domain, again because of the boundary conditions imposed there. Note that a constant free-stream temperature field is imposed such that no rms temperature profile exists.
The temporal evolution obtained by applying an averaging operator in the span-wise and lateral direction on the entrance velocity signals shows an exponential decay of the turbulence level similar to the spatial decay of grid-generated turbulence.
In order to extend the Reynolds number of the simulations, two successive spatial simulations were FIGURE 4 Temporal evolution of the turbulence levels.   The LESs were performed in the following way. For each domain only half of the box is considered as the last 16 cells in the longitudinal direction are progressively stretched before reaching the outflow boundary condition as one can observe in Fig. 5. For the calculation in the first domain ð0 # Re x # 43; 400Þ; we start with the laminar boundary layer solution. The calculation is initiated and the freestream turbulent field enters the domain. The interaction of the free-stream turbulence with the laminar boundary layer then takes place. After 2000 time steps (requiring input from two calculations of 1000 time steps to impose recorded entrance signals) we start to compute the statistics over 5000 time steps. During the last 1000 time steps, velocity and temperature signals are also recorded to be used as entrance signals for the second calculation domain ð43; 400 # Re x # 86; 800Þ: With this method, if the statistics at the exit of the first domain of calculation are not equivalent to those at the entrance of the second one, this implies that the flow was not statistically converged in the first domain. For the second simulation we proceed in the same manner except that now the initial field is the one obtained without free-stream turbulence for 43; 400 # Re x # 86; 800. Figures 6 and 7 show instantaneous velocity and temperature fields for case E ðTu < 10%Þ for Reynolds number varying from 0 ðx ¼ 0 mÞ up to 50,000 ðx < 0:9 mÞ: As observed earlier, the free-stream flow is maintained at a constant temperature. The formation of streak-like structures takes place rapidly after the leading edge for both fields. This supports the occurrence of bypass transition pointed out in Péneau et al. (2000).

RESULTS
In Figs. 8 and 9, ejection phenomena of equal intensity for both fields are observed.
Selective results representative of the evolution of the flow with the turbulence level are presented next. Figure 10 shows that up to Re x ¼ 73; 000 the mean longitudinal velocity profiles remain virtually insensitive to low turbulence level. With increasing turbulent level † If we disregard the statistics in the first cell of the domain where the no slip condition is progressively imposed.   (shown in Fig. 11), a progressive flattening of the velocity profile takes place inducing the formation of a wake. The wake formation was also observed experimentally by Dyban et al. (1977) and Dyban and Epick (1985) shown in Fig. 12. Figure 13 shows a more pronounced flattening of the temperature profile compared to the velocity profile. This was reported in Péneau et al. (2000) and is confirmed here with the finer grid simulations presented for both lower turbulent levels and higher Reynolds numbers. At higher turbulence levels we can observe in Fig. 14 that for case E, the profile is closer to a turbulent profile even though no logarithmic region is yet apparent.
The spatial evolution of the boundary layer thicknesses is plotted in Figs. 15 and 16. The absence of discontinuities around Re x < 40; 000 proves, at least up to Re x ¼ 80; 000; that the consecutive simulations methodology used does not perturb the spatial evolution. It also shows that at a moderate turbulent level, there is no significant thickening of the boundary layer in contrast with higher turbulence levels. The thickening explains the flattening of the velocity profile and, consequently, the reduction of the displacement thickness and increase of momentum thickness. The corresponding phenomena for the thermal field are even more important, hence the higher flattening of the temperature profile (Fig. 17) when the boundary layer thickness is used for normalizing the data.
The higher thickening of the thermal boundary layer is explained by the value of the turbulent Prandtl number inside the boundary layer. As shown in Figs. 18 and 19, whereas Pr t is close to 1 at low turbulence levels, Pr t , 1 over most of the boundary layer at a higher turbulence level, such as in case E. This implies the thermal turbulent structures transfer more energy and hence generate higher fluctuations in their wake than the dynamic structures.
Although at low turbulence levels no thickening of the boundary layer and no flattening of the velocity profile is observed, there are nevertheless some important phenomena that occur inside the boundary layer. Figures 20-22 show that a rapid increase of the maximum turbulence level takes place inside the boundary layer, and values exceeding free-stream levels are attained. This points to a mechanism of turbulent energy production inside the boundary layer. These observations are consistent with the experimental measurements of Dyban et al. (1977) (see Fig. 23).     The increase of u rms , and hence of Tu, seems to come about partly from a transfer from v rms inside the boundary layer. A reorganization of the turbulent structures in this region must be the origin of this phenomenon.
Examination of the turbulence level profile shows that peak Tu levels inside the boundary layer are about three times the outside turbulence level for all cases presented at Re x ¼ 73; 000: Since initially the free-stream turbulent fields are substantially different, this is unlikely to be a coincidence.
The analysis of the turbulent energy production profiles reveals that Tu max correspond to the location of maximum turbulent energy production. Based on wall variables, this production is of the same order as the fully turbulent boundary layer. The preceding figures show how important the history of the development of the boundary layer under the free-stream turbulent field is in the analysis.
For the thermal boundary layer, we note in Fig. 22 that the thermal turbulent level is equal or slightly higher than Tu for the same case and Reynolds number. Outside the thermal boundary layer, temperature fluctuations are still present over a distance of at least d T while the freestream thermal field is at constant temperature. This confirms the conclusion, drawn based on the turbulent Prandtl number, that the thermal boundary layer transfers more energy.
The analysis of the correlations kT 0 T 0 l; ku 0 T 0 l and kv 0 T 0 l inside the boundary layer reveals that the temperature fluctuations are aligned with v 0 . However, although up to the peak of Tu max the amplitude of T rms is directly correlated to u rms , after that point the correlation weakens until, in the wake of the boundary layer ku 0 T 0 l , kT 0 T 0 l , kv 0 T 0 l (see Figs. 24 and 25).
We deduce that the thermal fluctuations are convected by the v velocity field and this explains why the temperature fluctuations do not vanish outside the boundary layer. From this analysis we can conclude that the v velocity field feeds the thermal boundary layer in turbulent energy through the wake. kw 0 T 0 l is negligible throughout the computational domain. Figures 26 -29 show the impact of the above phenomena on wall shear stress and heat transfer coefficients, both of which are above the laminar values for Re x . 20; 000: This increase is linked to the amplitude of the free-stream turbulence level. For the case E, C f and St reach the turbulent value at Re x < 70; 000 or Re u < 200: For this case we note that wall coefficients do not change much for Re x . 20; 000; and as a first step to correlate wall transfer and free-stream turbulence a fit of the form ffiffiffiffiffiffiffi Re x p is proposed for C f /C f lam and St/St lam in Figs. 30 and 31.
Cases B and C in these figures exhibit the same increase of wall transfer and with no influence of the dissipative length scale. Furthermore, the temperature field is no more sensitive to free-stream turbulence than the velocity field since the same expressions fit the data. This appears to be in contradiction with the numerical and experimental observations. The answer to this contradiction lies in the rapid thickening of the boundary layers observed previously. Indeed, if instead of expressing the laminar value of C f and St in terms of Re x , we express them in FIGURE 24 Spatial evolution of the velocity temperature correlation for case E. -: Re x ¼ 16; 800; ----: Re x ¼ 33; 700; · · ·: Re x ¼ 55; 000; ----: Re x ¼ 71; 900: FIGURE 25 Spatial evolution of the velocity temperature correlation for case E. The same sign convention as in Fig. 24 is used -: Re x ¼ 16; 800; ----: Re x ¼ 33; 700; · · ·: Re x ¼ 55; 000; ----: Re x ¼ 71; 900: terms of d(x), namely C f ðxÞ ¼ ð3:32nÞ=ðU e dðxÞÞ and StðxÞ ¼ ð1:66DÞ=ðU e d T ðxÞÞ; we then obtain a completely different picture as shown in Figs. 32 -35, and comparison between Figs. 33 and 35 does indeed reveal the higher sensitivity of the thermal field to free-stream turbulence.
The evolution of C f /C f lam and St/St lam is also quite different as a linear variation is a good fit for the thermal field but a poor one for the dynamic field. The 500% increase for the Stanton number in Fig. 35 at Re x ¼ 60; 000 does not represent the actual enhancement since the comparison is between two different boundary layers of the same thickness with and without free-stream turbulence. Locally, the heat transfer increase is of the order of 200% as shown in Fig. 31. This illustrates nonetheless the fastest spatial development of the thermal boundary layer. It is reasonable to expect that transition to turbulence for the thermal boundary layer might occur earlier, spatially speaking, since the thicker a boundary layer, the more unstable it is.

CONCLUSIONS
The influence of free-stream turbulence on the development of a boundary layer was investigated using LES. The experimental observations of Dyban et al. (1977), Dyban and Epick (1985) and Bradshaw et al. and others are well reproduced by the simulations, and detailed analysis of the LES data provides an explanation for many of these observations. The simulations also underscore the higher sensitivity of the thermal field to free-stream turbulence.
For all five cases presented here, bypass transition is found to occur in the early stage of the boundary layer development. Even if all the characteristics of a fully  turbulent boundary layer are not yet observable, the high level of turbulent intensity as well as the important flattening of the mean velocity and temperature profile suggest that, at least for case E ðTu < 10%Þ; the transition to turbulence takes place at a relatively low Reynolds number. Simulations in progress appear to confirm this point.
The next phase of this work is to simulate the interaction of the free-stream turbulence with the boundary layer region up to a Reynolds number where fully turbulent flow occurs to allow comparison with data from ongoing parallel investigation at INRA by Kondjoyan.