Publications

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AIAA Journal

Fluctuating boundary layer wall shear stress can be an important loading component for structures subjected to turbulent boundary layer flows. While normal force loading via wall pressure fluctuation is relatively well described analytically, there is a dearth of information for wall shear behavior. Starting with an approximate acoustic analogy we derive simple approximate expressions for both wall pressure and wall shear fluctuations behavior utilizing a Taylor hypothesis based analogy between streamwise and temporal fluctuations. Analytical results include longitudinal spatial correlation, autocorrelation, frequency spectrum, RMS intensity and longitudinal and lateral coherence expressions. While coefficients in these expressions usually require some empirical input they nonetheless provide useful predictions for functional behavior. Comparison of the models with available literature data sets suggests reasonable agreement. Dedicated high fidelity numerical computations (direct numerical simulations) for a supersonic boundary layer are used to further explore the efficacy of these models. The analytical models for wall pressure fluctuation and wall shear fluctuation spectral density compare well for low frequency with the simulations when Reynolds number effects are included in the pressure fluctuation intensity. The approximate analytical models developed here provide a physics-based connection between classical empirical expressions and more complete experimental and computational descriptions.

Compressible jet-in-crossflow interactions are poorly simulated using Reynolds-Averaged Navier Stokes (RANS) equations. This is due to model-form errors (physical approximations) in RANS as well as the use of parameter values simply picked from literature (hence- forth, the nominal values of the parameters). Previous work on the Bayesian calibration of RANS models has yielded joint probability densities of C = (Cµ;Cϵ2;Cϵ1), the most influential parameters of the RANS equations. The calibrated values were far more predictive than the nominal parameter values and the advantage held across a range of freestream Mach numbers and jet strengths. In this work we perform Bayesian calibration across a range of Mach numbers and jet strengths and compare the joint densities, with a view of determining whether compressible jet-in-crossflow could be simulated with either a single joint probability density or a point estimate for C. We find that probability densities for ;Cϵ2 agree and also indicate that the range typically used in aerodynamic simulations should be extended. The densities for ;Cϵ1 agree, approximately, with the nominal value. The densities for ;Cµ do not show any clear trend, indicating that they are not strongly constrained by the calibration observables, and in turn, do not affect them much. We also compare the calibrated results to a recently developed analytical model of a jet-in-cross flow interaction. We find that the values of C estimated by the analytical model delivers prediction accuracies comparable to the calibrated joint densities of the parameters across a range of Mach numbers and jet strengths.

The k-ε turbulence model has been described as perhaps “the most widely used complete turbulence model.” This family of heuristic Reynolds Averaged Navier-Stokes (RANS) turbulence closures is supported by a suite of model parameters that have been estimated by demanding the satisfaction of well-established canonical flows such as homogeneous shear flow, log-law behavior, etc. While this procedure does yield a set of so-called nominal parameters, it is abundantly clear that they do not provide a universally satisfactory turbulence model that is capable of simulating complex flows. Recent work on the Bayesian calibration of the k-ε model using jet-in-crossflow wind tunnel data has yielded parameter estimates that are far more predictive than nominal parameter values. Here we develop a self-similar asymptotic solution for axisymmetric jet-in-crossflow interactions and derive analytical estimates of the parameters that were inferred using Bayesian calibration. The self-similar method utilizes a near field approach to estimate the turbulence model parameters while retaining the classical far-field scaling to model flow field quantities. Our parameter values are seen to be far more predictive than the nominal values, as checked using RANS simulations and experimental measurements. They are also closer to the Bayesian estimates than the nominal parameters. A traditional simplified jet trajectory model is explicitly related to the turbulence model parameters and is shown to yield good agreement with measurement when utilizing the analytical derived turbulence model coefficients. The close agreement between the turbulence model coefficients obtained via Bayesian calibration and the analytically estimated coefficients derived in this paper is consistent with the contention that the Bayesian calibration approach is firmly rooted in the underlying physical description.

The development of the unsteady pressure field on the floor of a rectangular cavity was studied at Mach 0.9 using high-frequency pressure-sensitive paint. Power spectral amplitudes at each cavity resonance exhibit a spatial distribution with an oscillatory pattern; additional maxima and minima appear as the mode number is increased. This spatial distribution also appears in the propagation velocity of modal pressure disturbances. This behavior was tied to the superposition of a downstream-propagating shear-layer disturbance and an upstream-propagating acoustic wave of different amplitudes and convection velocities, consistent with the classical Rossiter model. The summation of these waves generates an interference pattern in the spatial pressure amplitudes and resulting phase velocity of the resonant pressure fluctuations.

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Reynolds-Averaged Navier-Stokes models are not very accurate for high-Reynolds-number compressible jet-incrossflow interactions. The inaccuracy arises from the use of inappropriate model parameters and model-form errors in the Reynolds-Averaged Navier-Stokes model. In this work, the hypothesis is pursued that Reynolds-Averaged Navier-Stokes predictions can be significantly improved by using parameters inferred from experimental measurements of a supersonic jet interacting with a transonic crossflow.ABayesian inverse problem is formulated to estimate three Reynolds-Averaged Navier-Stokes parameters (Cμ;Cϵ2;Cϵ1), and a Markov chain Monte Carlo method is used to develop a probability density function for them. The cost of the Markov chain Monte Carlo is addressed by developing statistical surrogates for the Reynolds-Averaged Navier-Stokes model. It is found that only a subset of the (Cμ;Cϵ2;Cϵ1) spaceRsupports realistic flow simulations.Ris used as a prior belief when formulating the inverse problem. It is enforced with a classifier in the current Markov chain Monte Carlo solution. It is found that the calibrated parameters improve predictions of the entire flowfield substantially when compared to the nominal/ literature values of (Cμ;Cϵ2;Cϵ1); furthermore, this improvement is seen to hold for interactions at other Mach numbers and jet strengths for which the experimental data are available to provide a comparison. The residual error is quantifies, which is an approximation of the model-form error; it is most easily measured in terms of turbulent stresses.

Here we provide a discussion regarding the applicability of a family of traditional heat transfer correlation based models for several (unit level) heat transfer problems associated with flight heat transfer estimates and internal flow heat transfer associated with an experimental simulation design (Dobranich 2014). Variability between semi-empirical free-flight models suggests relative differences for heat transfer coefficients on the order of 10%, while the internal annular flow behavior is larger with differences on the order of 20%. We emphasize that these expressions are strictly valid only for the geometries they have been derived for e.g. the fully developed annular flow or simple external flow problems. Though, the application of flat plate skin friction estimate to cylindrical bodies is a traditional procedure to estimate skin friction and heat transfer, an over-prediction bias is often observed using these approximations for missile type bodies. As a correction for this over-estimate trend, we discuss a simple scaling reduction factor for flat plate turbulent skin friction and heat transfer solutions (correlations) applied to blunt bodies of revolution at zero angle of attack. The method estimates the ratio between axisymmetric and 2-d stagnation point heat transfer skin friction and Stanton number solution expressions for sub-turbulent Reynolds numbers %3C1x10 4 . This factor is assumed to also directly influence the flat plate results applied to the cylindrical portion of the flow and the flat plate correlations are modified by

Here we derive an approximate turbulent self-similar model for a class of favorable pressure gradient wedge-like flows, focusing on the stagnation point limit. While the self-similar model provides a useful gross flow field estimate this approach must be combined with a near wall model is to determine skin friction and by Reynolds analogy the heat transfer coefficient. The combined approach is developed in detail for the stagnation point flow problem where turbulent skin friction and Nusselt number results are obtained. Comparison to the classical Van Driest (1958) result suggests overall reasonable agreement. Though the model is only valid near the stagnation region of cylinders and spheres it nonetheless provides a reasonable model for overall cylinder and sphere heat transfer. The enhancement effect of free stream turbulence upon the laminar flow is used to derive a similar expression which is valid for turbulent flow. Examination of free stream enhanced laminar flow suggests that the rather than enhancement of a laminar flow behavior free stream disturbance results in early transition to turbulent stagnation point behavior. Excellent agreement is shown between enhanced laminar flow and turbulent flow behavior for high levels, e.g. 5% of free stream turbulence. Finally the blunt body turbulent stagnation results are shown to provide realistic heat transfer results for turbulent jet impingement problems.

Turbulent fluctuation behavior is approximately modeled using a pulsatile flow model analogy.. This model follows as an extension to the turbulent laminar sublayer model developed by Sternberg (1962) to be valid for a fully turbulent flow domain. Here unsteady turbulent behavior is modeled via a sinusoidal pulsatile approach. While the individual modes of the turbulent flow fluctuation behavior are rather crudely modeled, approximate temporal integration yields plausible estimates for Root Mean Square (RMS) velocity fluctuations. RMS pressure fluctuations and spectra are of particular interest and are estimated via the pressure Poisson expression. Both RMS and Power Spectral Density (PSD), i.e. spectra are developed. Comparison with available measurements suggests reasonable agreement. An additional fluctuating quantity, i.e. RMS wall shear fluctuation is also estimated, yielding reasonable agreement with measurement.

We examine the role of periodic sinusoidal free-stream disturbances on the inner law law-of-the-wall (log-law) for turbulent boundary layers. This model serves a surrogate for the interaction of flight vehicles with atmospheric disturbances. The approximate skin friction expression that is derived suggests that free-stream disturbances can cause enhancement of the mean skin friction. Considering the influence of grid generated free stream turbulence in the laminar sublayer/log law region (small scale/high frequency) the model recovers the well-known shear layer enhancement suggesting an overall validity for the approach. The effect on the wall shear associated with the lower frequency due to the passage of the vehicle through large (vehicle scale) atmospheric disturbances is likely small i.e. on the order 1% increase for turbulence intensities on the order of 2%. The increase in wall pressure fluctuation which is directly proportional to the wall shear stress is correspondingly small.

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We demonstrate a Bayesian method that can be used to calibrate computationally expensive 3D RANS models with complex response surfaces. Such calibrations, conditioned on experimental data, can yield turbulence model parameters as probability density functions (PDF), concisely capturing the uncertainty in the estimation. Methods such as Markov chain Monte Carlo construct the PDF by sampling, and consequently a quick-running surrogate is used instead of the RANS simulator. The surrogate can be very difficult to design if the model’s response i.e., the dependence of the calibration variable (the observable) on the parameters being estimated is complex. We show how the training data used to construct the surrogate models can also be employed to isolate a promising and physically realistic part of the parameter space, within which the response is well-behaved and easily modeled. We design a classifier, based on treed linear models, to model the “well-behaved region”. This classifier serves as a prior in a Bayesian calibration study aimed at estimating 3 k-ε parameters C = (Cμ, Cε2, Cε1) from experimental data of a transonic jet-in-crossflow interaction. The robustness of the calibration is investigated by checking its predictions of variables not included in the calibration data. We also check the limit of applicability of the calibration by testing at an off-calibration point.