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Turbulent Boundary-Layer
A stochastic model for the space-time turbulent boundary-layer wall-pressure spectrum was developed that uses statistical data from Reynolds-Averaged Navier-Stokes (RANS) solutions as input. The model integrates the source terms for the surface-pressure covariance across the boundary layer for user-specified space and time separations to form a discrete surface-pressure correlation function, the Fourier transform of which yields the surface-pressure wave number-frequency spectrum. By integrating RANS data into the model, it is able to respond to local geometry and flow conditions. Validation cases show that predicted surface-pressure power spectra respond appropriately to favorable, zero, and adverse pressure gradients. By operating as a post-processor of CFD RANS analyses, the model is a predictive tool that can be used in flow and flow-induced noise analyses. Because contemporary RANS models are able to predict flow statistics well for configurations of practical interest, this approach to modeling the turbulent boundary-layer forcing function is expected to generalize well to new flow configurations without requiring flow-specific tuning.

The bulk of research on the behavior of the velocity and pressure fluctuations within turbulent boundary layer (TBL) flows has been conducted under ideal conditions, such as on flows with zero pressure gradient over flat plates (Klebanoff, 1954), and inside cylinder (pipe) interiors (Laufer, 1954) and channels (Laufer, 1950). Estimates have been made by many authors of the wall pressure autospectra and cross-spectra for a variety of flow speeds and fluid properties. The survey by Bull (1996) summarizes much of the existing research. The wall pressure autospectra may be collapsed reasonably well using combinations of so-called inner or outer flow variables, where outer variable scaling collapses low-frequency levels well, and inner variable scaling collapses high-frequency levels well. Keith, Hurdis, and Abraham (1991) explored mixed variable scaling, where the frequency is scaled on outer variables and the pressure levels scaled on inner variables, and obtained good general collapse of levels over all frequencies.

Empirical models of wall pressure autospectra under zero pressure gradient TBL flow have been proposed by several authors, such as Corcos (1963), Chase (1980, 1987), and Smolyakov and Tkachenko (1991) that are essentially curve fits to the scaled measured datasets. Figure 1 shows the model of Smolyakov and Tkachenko plotted against several sets of measured data compiled by Keith, Hurdis, and Abraham using mixed variable scaling. Since all pressure sensors attenuate high-wavenumber fluctuations over their surfaces to some degree, the well known Corcos correction (1963) is applied to the measurements shown in Figure 1.

Gavin (2002) present two empirical models tuned to fit his data. One, the Simplified Anisotropic Model (SAM), is adopted as the turbulence velocity-correlation closure for (11). Gavin’s measurements show that a turbulence-velocity correlation volume can be modeled as an ellipsoid inclined at an angle q to the wall (see Figure 2). From his measurements, he extracts estimates of the inclination angle and of the stretching relationships between the major and minor axes of the ellipsoid.

In Figure 3, we present separation-time contours of the SAM-modeled correlation functions, and contours for spatial separations in Figure 4. These data reproduce the data presented by Gavin. Of note, the correlation volume in time and in the streamwise direction is large for the C11 component relative to the other components. Both C11 and C33 are elongated in time and the streamwise direction.

To generate RANS data similar to Schloemer’s results, a numerical experiment is constructed that provides adverse, approximately-zero, and favorable pressure-gradient regions. A schematic of the RANS domain is shown in Figure 5.

Mean-velocity profiles are presented in Figure 6. The solid lines are from RANS and Schloemer’s data are the symbols. The agreement is good. The largest deviation is for the zero-pressure-gradient case, consistent with the reported difference in the first shape factor. The mean-velocity data in wall coordinates are presented in Figure 6b. The zero pressure gradient case follows the universal profile and shows a well developed logarithmic region. The adverse- and favorable-pressure gradient cases are close to the universal curve for the inner region of the boundary layer but deviate in the outer flow, where the adverse pressure gradient exceeds the zero pressure-gradient one while the favorable pressure-gradient profile undershoots the zero-pressure gradient result. Turbulence data from RANS are presented in Figure 7. The dimensionless turbulence length scale, Figure 7a, for the q-w model is defined as , where , the proportionality constant, is observed to be between 0.54 and 0.65 (0.54 is used for the current model). The dimensionless turbulence length scale shows an increasing trend from adverse to favorable pressure gradient. The data are nondimensionalized on the boundary-layer thickness, so the larger value for favorable pressure gradient reflects the thinning boundary layer for that case. The turbulence length scale for the adverse pressure gradient is physically longer, but the boundary-layer thickness is comparatively larger too. Profiles of the turbulence kinetic energy (tke) are presented in Figure 7b. Schloemer presents data for the streamwise root-mean-square (urms) value of velocity. The square of that statistic provides one component of the tke. Integrand and cumulative results for the surface-pressure variance are presented in Figure 8, nondimensionalized by inner variables. The integrand profiles in Figure 8 suggest the existence of two separated contributions, one from the inner layer and a second from the outer layer. The surface-pressure covariances for the favorable, zero, and adverse pressure gradient cases are presented in Figure 9. The color scheme is consistent with Figure 8. Components of the surface-pressure spectra are presented in Figure 10. Neither the TTM nor the TTS contributions are significant. The primary contribution is from TMS interactions for all cases. The TTN contribution has an increasing importance with decreasing pressure gradient. The total spectra are replotted in Figure 11 with Schloemer’s data overlayed as symbols. The low-frequency levels of the predicted spectra agree well with Schloemer.

Images
View Image (20kb)	Figure 1: Smolyakov and Tkachenko TBL wall-pressure autospectra model vs. measured data
View Image (15kb)	Figure 2: Gavin’s mapping of an ellipsoidal correlation volume onto a spherical one.
View Image (36kb)	Figure 3: Contours of Gavin’s SAM C11, C22, and C33 correlation function model for separations in the wall-normal coordinate direction, y, and time, t.
View Image (57kb)	Figure 4: Vertical profiles and contours in the wall-normal, y, and cross-stream, z, coordinate directions of Gavin’s SAM C11, C22, and C33 correlation function models
View Image (18kb)	Figure 5: Computational domain showing the approximate stations for extracting favorable (FPG), zero (ZPG), and adverse (APG) pressure-gradient RANS data (colored by static pressure).
View Image (25kb)	Figure 6: Mean-velocity profiles a) compared to Schloemer’s data and b) in wall coordinates.
View Image (25kb)	Figure 7: Profiles of the turbulence correlation length () and kinetic energy (k) from RANS: top panels normalized by the boundary layer depth (d) and the velocity at d (ue) and the bottom panels in wall units.
View Image (21kb)	Figure 8: Integrand (solid) and cumulative (dashed) contributions to the surface-pressure variance in wall coordinates: TMS (red), TTN (green), TTM (blue), and TTS(orange).
View Image (17kb)	Figure 9: The surface-pressure correlation function vs. separation time for favorable, zero, and adverse pressure-gradient conditions: TMS (red), TTN (green), TTM (blue), and TTS(orange).
View Image (17kb)	Figure 10: The surface-pressure spectra for favorable, zero, and adverse pressure-gradient conditions: TMS (red), TTN (green), TTM (blue), TTS(orange), and TMS+TTN (black).
View Image (31kb)	Figure 11: Predicted surface-pressure spectra (lines) compared to a) Schloemer’s data (symbols) for favorable (red), zero (green), and adverse (blue) pressure gradients and b) the Bull &Thomas, Farabee, Schloemer, Willmarth & Woodrigde, and Blakewell zero pressure gradient datasets.