Chapter 1
DNS of Turbulent Boundary Layers Subjected to Adverse Pressure Gradients Guillermo Araya and Luciano Castillo
Abstract Direct Numerical Simulations (DNS) of spatially-developing turbulent boundary layers with prescribed moderate and strong adverse pressure (APG) gradients are performed. A method for prescribing realistic turbulent velocity inflow boundary conditions is employed based on the on the dynamic multi-scale approach proposed by [1] [2]; and, it is an extension of the rescaling-recycling method by [6]. Comparison with data from more costly DNS ([7][5]) yields accurate results. In addition, the dynamic multi-scale approach does not require lengthy computational domains as in [7] and [5]. Furthermore, it is shown that in APG flows the presence of a second outer peak in u′+ rms is more pronounce than in ZPG flows. Additionally, the plateau between the inner and outer peaks suggests the presence of an overlap (i.e., meso-layer) in the mean velocity profile, as discussed in GC-97 [3]. Moreover, these outer peaks are also observed in the production of turbulence even at low Reynolds numbers. Finally, the mean velocity profiles in wall-units show that the wake region is magnified as the APG strengths increases. This suggests that the large scales in the outer flow dominate most of the boundary layer.
1.1 Numerical results in APG flows For simulations of moderate and strong APG, the curvature at the upper surface of the computational box is prescribed in such a way to obtain a power-law variation of the freestream velocity, i.e. U∞ ∼ (x − xo )m with m = -0.17 and -0.22. Table 1.1 summarizes the different APG cases with the corresponding Reθ , domain dimenGuillermo Araya Civil & Computational Engineering Centre, Swansea University, Swansea, UK, e-mail:
[email protected] Luciano Castillo Dept. of Mechanical, Aeronautical and Nuclear Eng., Rensselaer Polytechnic Institute, Troy, USA e-mail:
[email protected]
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Guillermo Araya and Luciano Castillo
sions, number of mesh points, mesh resolutions, time step, the collected sample in viscous time scales and the prescribed inlet boundary layer thicknesses.
Table 1.1: Table showing proposed DNS cases and domain parameters for APG flows. APG
Parameter
Moderate Strong Reθ 438-633 1029-1512 Lx /δinl 15 12 Ly /δinl 3.8 3 Lz /δinl 1.6 3.2 Nx Ny Nz 120 × 65 × 50 150 × 90 × 100 ∆ x+ 20 21 + ∆ y+ 0.2/6.8 0.2/17 min /∆ ymax ∆ z+ 5 8 ∆t+ 0.59 0.22 Tsample + δinl
u2τ ν
1770 161
1540 271
′+ ′+ The turbulence intensities (u′+ rms , vrms and wrms ) and Reynolds shear stresses + (−u′ v′ ) are depicted in Fig. 1.1(a), at Reθ = 580 and moderate APG. The pressure gradient effects are mostly manifested as outer peaks in the outer region around y+ ≈ 100. This phenomenon is more evident in the streamwise velocity fluctuations, u′+ rms , of APG than those of ZPG flows ([4]). As the adverse pressure gradi+ ent becomes stronger, the outer peak in u′+ rms at y ≈ 200 in Fig 1.1(b) is almost as high as the inner peak at y+ ≈ 10. Notice that maxima of the Reynolds shear + stresses (−u′ v′ ), shown in Figs. 1.1(a) and 1.2(a), move away from the wall as the adverse pressure gradient strength increases. It is also observed that the Reynolds shear stress profiles have slopes given by the pressure gradient as expected, which becomes very steep for the strong APG case. In addition, the agreement of present velocity fluctuations with simulations by [7] and [5] at similar Reynolds numbers (Reθ ) and APG strengths (m) is quite good. The observed agreement is particularly encouraging considering that the streamwise domain lengths of these prior studies by [7] and [5] were 4 and 14 times larger, respectively, than those of presently reported simulations using the dynamic multi-scale method. Furthermore, the effects of the APG strengths are also appreciated in the main term of the turbulence production (see fig. 1.2(b)) as peaks in the outer layer due to the influence of large scales in this zone. Note the excellent agreement of present turbulence production at moderate APG with the more costly DNS results from [7]. Figure 1.3 exhibits the mean velocities together with the streamwise turbulence intensities for both APG cases in wall units. One can observe the highly pronounced wake regions in U + as the APG strength increases. Notice that the regions between the inner and outer peaks of u′+ rms are probably overlap regions mainly characterized by a meso-layer due to the low Reθ and not an inertia sub-layer or the log-layer as stated in [4] for much higher
1 DNS of Turbulent Boundary Layers Subjected to Adverse Pressure Gradients
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Reθ , where a plateau in the normal Reynolds stresses exists. This meso-layer is very important since a direct interaction between the small and large scales exists. According to fig. 1 in [4], these outer peaks are also observed in the energy spectra at length scales that usually are several times larger than the boundary layer thickness (≈ 6δ ). In ZPG flows the second peak in u′+ rms occurs at much higher Reθ than those in APG flows. For instance, in [4] the Reθ ≈ 21, 000 whereas in present simulations is in the order of 1,500.
Present DNS m=-0.17 Reθ=580 Skote (2001) DNS m=-0.15 Reθ=685
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Present DNS m=-0.22 Reθ=1370 Present DNS m=-0.22 Reθ=1460 Lee and Sung (2008) DNS m=-0.20 Reθ=1350
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Fig. 1.1: Turbulence intensities and Reynolds shear stresses of moderate APG (a) and turbulence intensities of strong APG (b).
Present DNS m=-0.22 Reθ=1370 Present DNS m=-0.22 Reθ=1460 Lee and Sung (2008) DNS m=-0.20 Reθ=1350
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Present DNS - ZPG - Reθ=380 Present DNS - Moderate APG - Reθ=620 Present DNS - Strong APG - Reθ=1460 Skote (2001) DNS - Moderate APG - Reθ=685 ______ + + +
0.5 0.45
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Fig. 1.2: Reynolds shear stresses of strong APG (a) and turbulence production in ZPG and APG flows (b).
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Guillermo Araya and Luciano Castillo 30
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Present DNS m=-0.17 Reθ=580
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Present DNS m=-0.22 Reθ=1460
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Fig. 1.3: Mean velocity and streamwise turbulence intensity profiles in wall units for moderate (a) and strong (b) APG.
1.2 Final remarks A generalized dynamic multi-scale method has been proposed and evaluated as turbulent inflow generator in spatially-developing boundary layers subjected to APG. The effects of adverse pressure gradients on turbulent boundary layers are mainly manifested in the outer region, as a second local peak; particularly, in profiles of u′+ rms even at low Reynolds numbers. This clearly provokes high levels of production in the outer region. Additionally, a plateau is observed between the inner and outer peaks of u′+ rms and it demonstrates the presence of an overlap (i.e., meso-layer) in the mean velocity profiles, as described in GC-97 [3] for low Reynolds numbers data.
References 1. Araya, G., Jansen, K. & Castillo, L. 2009 Inlet condition generation for spatially-developing turbulent boundary layers via multi-scale similarity. Journal of Turbulence, 10, No. 36, 1–33. 2. Araya G., Castillo L., Meneveau C. & Jansen K. 2011 A dynamic multi-scale approach for turbulent inflow boundary conditions in spatially-developing flows. Journal of Fluid Mechanics, doi:10.1017/S0022112010005616. 3. George, W.K. & Castillo, L. 1997 Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689-729. 4. Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Philosophical Transactions of The Royal Society A 365, 647–664. 5. Lee, J. & and Sung, H. 2008 Effects of an adverse pressure gradient on a turbulent boundary layer. Int. J. of Heat and Fluid Flow 29 (3), 568–578. 6. Lund, T.S., Wu, X. & Squires, K.D. 1998 Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations. J. Comp. Phys 140, 233–258. 7. Skote, M. 2001 Studies of turbulent boundary layer flow through Direct Numerical Simulation. PhD thesis, Royal Institute of Technology, Stockholm, Sweden.
