Here we examine models for particle curtain dispersion using drag based formalisms and their connection to streamwise pressure difference closures. Focusing on drag models, we specifically demonstrate that scaling arguments developed in DeMauro et. al. [1] using early time drag modeling can be extended to include late time particle curtain dispersion behavior by weighting the dynamic portion of the drag relative velocity e.g. (Formula Presented) by the inverse of the particle volume fraction to the ¼th power. The additional parameter e.g. α introduced in this scaling is related to the model drag parameters by employing an early-time latetime matching argument. Comparison with the scaled measurements of DeMauro et. al. suggest that the proposed modification is an effective formalism. Next, the connection between drag-based models and streamwise pressure difference-based expressions is explored by formulating simple analytical models that verify an empirical (Daniel and Wagner [2]) upstream-downstream expression. Though simple, these models provide physics-based approached describing shock particle curtain interaction behavior.
Here we consider the shock stand-off distance for blunt forebodies using a simplified differential-based approach with extensions for high enthalpy dissociative chemistry effects. Following Rasmussen [4], self-similar differential equations valid for spherical and cylindrical geometries that are modified to focus on the shock curvature induced vorticity in the immediate region of the shock are solved to provide a calorically perfect estimate for shock standoff distance that yields good agreement with classical theory. While useful as a limiting case, strong shock (high enthalpy) calorically perfect results required modification to include the effects of dissociative thermo-chemistry. Using a dissociative ideal gas model for dissociative equilibrium behavior combined with shock Hugoniot constraints we solve to provide thermodynamic modifications to the shock density jump thereby sensitizing the simpler result for high enthalpy effects. The resulting estimates are then compared to high enthalpy stand-off data from literature, recent dedicated high speed shock tunnel measurements and multi-temperature partitioned implementation CFD data sets. Generally, the theoretical results derived here compared well with these data sources, suggesting that the current formulation provides an approximate but useful estimate for shock stand-off distance.
The near wake flow field associated with hypersonic blunt bodies is characterized by complex physical phenomena resulting in both steady and time dependent pressure loadings on the base of the vehicle. Here, we focus on the unsteady fluid dynamic pressure fluctuation behavior as a vibratory input loading. Typically, these flows are characterized by a locally low-pressure, separated flow region with an unsteady formation of vortical cells that are locally produced and convected downstream into the far-field wake. This periodic production and transport of vortical elements is very-well known from classical incompressible fluid mechanics and is usually termed as the (Von) Karman vortex street. While traditionally discussed within the scope of incompressible flow, the periodic vortex shedding phenomenon is known for compressible flows as well. To support vehicle vibratory loading design computations, we examine a suite of analytical and high-fidelity computational models supported by dedicated experimental measurements. While large scale simulation approaches offer very high-quality results, they are impractical for design-level decisions, implying that analytically derived reduced order models are essential. The major portions of this effort include an examination of the DeChant-Smith Power Spectral Density (PSD) [1] model to better understand both overall Root Mean Square (RMS) magnitude and functional maximum associated with a critical vortex shedding phenomenon. The critical frequency is examined using computational, experiments and an analytical shear layer frequency model. Finally, the PSD magnitude maximum is studied using a theory-based approach connecting the PSD to the spatial correlation that strongly supports the DeChant-Smith PSD model behavior. These results combine to demonstrate that the current employed PSD models provide plausible reduced order closures for turbulent base pressure fluctuations for high Reynolds number flows over range of Mach numbers. Access to a reliable base pressure fluctuation model then permits simulation of bluff body vibratory input.
High-enthalpy hypersonic flight represents an application space of significant concern within the current national-security landscape. The hypersonic environment is characterized by high-speed compressible fluid mechanics and complex reacting flow physics, which may present both thermal and chemical nonequilibrium effects. We report on the results of a three-year LDRD effort, funded by the Engineering Sciences Research Foundation (ESRF) investment area, which has been focused on the development and deployment of new high-speed thermochemical diagnostics capabilities for measurements in the high-enthalpy hypersonic environment posed by Sandia's free-piston shock tunnel. The project has additionally sponsored model development efforts, which have added thermal nonequilibrium modeling capabilities to Sandia codes for subsequent design of many of our shock-tunnel experiments. We have cultivated high-speed, chemically specific, laser-diagnostic approaches that are uniquely co-located with Sandia's high-enthalpy hypersonic test facilities. These tools include picosecond and nanosecond coherent anti-Stokes Raman scattering at 100-kHz rates for time-resolved thermometry, including thermal nonequilibrium conditions, and 100-kHz planar laser-induced fluorescence of nitric oxide for chemically specific imaging and velocimetry. Key results from this LDRD project have been documented in a number of journal submissions and conference proceedings, which are cited here. The body of this report is, therefore, concise and summarizes the key results of the project. The reader is directed toward these reference materials and appendices for more detailed discussions of the project results and findings.
Here we examine approximate geometrically self-similar solutions to a parabolic free boundary value problem applied to alluvial fan surface morphology and growth. Alluvial fans are fan- or cone-shaped sedimentary deposits caused by the rapid deposition of sediment from a canyon discharging onto a flatter plain. Longitudinal, topographic profiles of fans can be readily described by a seemingly time independent dimensionless profile (DeChant et al., 1999). However, because an alluvial fan can be expected to grow over time, it is clear that this “steady” profile is certainly time dependent and can be described using a space-time self-similar solution. In an experimental and theory-based study, Guerit et al. (2014) developed a self-similar (or as they describe it a self-affine) linear solution based upon an approximate first order small parameter expansion solution for a 1-d homogeneous nonlinear diffusion equation. Direct substitution of this result into a linear diffusion equation suggests that this first order expression may not fully satisfy the associated governing equation. In contrast, we develop a more complete solution based upon a modeled approximation for the axi-symmetric formulation such that the associated temporal behavior is consistent with a 1/3 time power-law as described by Reitz and Jerolmack (2014). The resulting expression is an exact solution to a linear heat equation. We emphasize that a small parameter is not inherent to the resulting profile result and is not included in our model development. Though developed using rather different approaches, the formal solution developed here is in good agreement with the simple polynomial described by DeChant et al. (1999) suggesting that this self-similar solution is a suitable time dependent representation of alluvial fan longitudinal profile form and improves on earlier work.
Fluctuating boundary layer pressure fluctuations are an important loading component for reentry bodies. Characterization of these loads is often described through cross-spectral density-based definitions, such as, longitudinal and lateral coherence, spatial correlation and frequency power spectral density. The widely utilized Corcos separable coherence model functional form has been employed in this study. While the classical Corcos D xD style model using a self-similar velocity-spacing variable e.g. (here the subscript denotes a dimensional U vaiable) has been effectively used for low speed simulations, high speed problems often require a model that involves both the self-similar variable and the sensor spacing D Here we examine longitudinal coherence formulations that include explicit D behavior as well as the self-similar variable. Examination of an analytical model/synthetic pressure fluctuation correlation function developed here clearly demonstrate that the self-similar form may need to be supplement by non-similar information. Using the synthetic space-time correlation expression, a coherence model which uses self-similar variables and explicit (but continuous) spatial information is proposed. Estimates for the parameters in the coherence model are derived using asymptotic arguments available from the synthetic result. Further, relationships are derived to estimate coherence model parameters and their connection to longitudinal correlation behavior assuming exponential auto-spectral density models. Comparison of these expressions with wind tunnel test and DNS simulation shows good comparison. Measurements from flight tests which deviate greatly from the classical self-similar form can be successfully described using the extended model although the coherence model parameters must be modified. In summary, an extended coherence model is developed which provides good explanations of longitudinal coherence and correlation behavior.
Fluctuating boundary layer pressure fluctuations are an important loading component for reentry bodies. Characterization of these loads is often described through cross-spectral density-based definitions, such as, longitudinal and lateral coherence, spatial correlation and frequency power spectral density. The widely utilized Corcos separable coherence model functional form has been employed in this study. While the classical Corcos D xD style model using a self-similar velocity-spacing variable e.g. (here the subscript denotes a dimensional U vaiable) has been effectively used for low speed simulations, high speed problems often require a model that involves both the self-similar variable and the sensor spacing D Here we examine longitudinal coherence formulations that include explicit D behavior as well as the self-similar variable. Examination of an analytical model/synthetic pressure fluctuation correlation function developed here clearly demonstrate that the self-similar form may need to be supplement by non-similar information. Using the synthetic space-time correlation expression, a coherence model which uses self-similar variables and explicit (but continuous) spatial information is proposed. Estimates for the parameters in the coherence model are derived using asymptotic arguments available from the synthetic result. Further, relationships are derived to estimate coherence model parameters and their connection to longitudinal correlation behavior assuming exponential auto-spectral density models. Comparison of these expressions with wind tunnel test and DNS simulation shows good comparison. Measurements from flight tests which deviate greatly from the classical self-similar form can be successfully described using the extended model although the coherence model parameters must be modified. In summary, an extended coherence model is developed which provides good explanations of longitudinal coherence and correlation behavior.
[Abstract] To support reduced order modeling of heat transfer for reentry bodies we develop an approximate solution method is identified that provides good estimates for the local wall derivative (and thereby the skin friction and Nusselt numbers) for a wide range of self-similar laminar formulations. These formulations include: Blasius flow, axisymmetric and planar stagnation flows and the Faulkner-Skan flows. The approach utilized is simply an extension of the classical Weyl formulation for the Blasius equation. Using this solution form estimates that naturally represent combined flow behaviors are represented without post-solution interpolation. An important example, namely axisymmetric stagnation equally combined with laminar zero pressure gradient (flat plate) flow, shows a difference of 10% between the pre-solution combination developed here and s simple post-solution arithmetic average. Clearly, the nonlinearity inherent to these solutions prevails in terms of these simple solutions. Compressible extensions to the basic incompressible result are achieved by including a near wall Chapman-Rubesin term making these solutions suitable for adiabatic wall problems. Direct comparison of the wall gradient estimation procedure developed here demonstrates excellent agreement with empirically fit blunt body heat transfer models such as the asymptotically consistent model of Kemp et. al. which are deemed more appropriate than the classical stagnation point scaling approaches.
Here, experiments were performed within Sandia National Labs’ Multiphase Shock Tube to measure and quantify the shock-induced dispersal of a shock/dense particle curtain interaction. Following interaction with a planar travelling shock wave, schlieren imaging at 75 kHz was used to track the upstream and downstream edges of the curtain. Data were obtained for two particle diameter ranges ($d_{p}=106{-}125$,$300{-}355~\unicode[STIX]{x03BC}\text{m}$) across Mach numbers ranging from 1.24 to 2.02. Using these data, along with data compiled from the literature, the dispersion of a dense curtain was studied for multiple Mach numbers (1.2–2.6), particle sizes ($100{-}1000~\unicode[STIX]{x03BC}\text{m}$) and volume fractions (9–32 %). Data were non-dimensionalized according to two different scaling methods found within the literature, with time scales defined based on either particle propagation time or pressure ratio across a reflected shock. The data refelct that spreading of the particle curtain is a function of the volume fraction, with the effectiveness of each time scale based on the proximity of a given curtain’s volume fraction to the dilute mixture regime. It is observed that volume fraction corrections applied to a traditional particle propagation time scale result in the best collapse of the data between the two time scales tested here. In addition, a constant-thickness regime has been identified, which has not been noted within previous literature.
Compressible jet-in-crossflow interactions are difficult to simulate accurately using Reynolds-averaged Navier-Stokes (RANS) models. This could be due to simplifications inherent in RANS or the use of inappropriate RANS constants estimated by fitting to experiments of simple or canonical flows. Our previous work on Bayesian calibration of a k - ϵ model to experimental data had led to a weak hypothesis that inaccurate simulations could be due to inappropriate constants more than model-form inadequacies of RANS. In this work, Bayesian calibration of k - ϵ constants to a set of experiments that span a range of Mach numbers and jet strengths has been performed. The variation of the calibrated constants has been checked to assess the degree to which parametric estimates compensate for RANS's model-form errors. An analytical model of jet-in-crossflow interactions has also been developed, and estimates of k - ϵ constants that are free of any conflation of parametric and RANS's model-form uncertainties have been obtained. It has been found that the analytical k - ϵ constants provide mean-flow predictions that are similar to those provided by the calibrated constants. Further, both of them provide predictions that are far closer to experimental measurements than those computed using "nominal" values of these constants simply obtained from the literature. It can be concluded that the lack of predictive skill of RANS jet-in-crossflow simulations is mostly due to parametric inadequacies, and our analytical estimates may provide a simple way of obtaining predictive compressible jet-in-crossflow simulations.
We demonstrate a statistical procedure for learning a high-order eddy viscosity model (EVM) from experimental data and using it to improve the predictive skill of a Reynoldsaveraged Navier-Stokes (RANS) simulator. The method is tested in a three-dimensional (3D), transonic jet-in-crossflow (JIC) configuration. The process starts with a cubic eddy viscosity model (CEVM) developed for incompressible flows. It is fitted to limited experimental JIC data using shrinkage regression. The shrinkage process removes all the terms from the model, except an intercept, a linear term, and a quadratic one involving the square of the vorticity. The shrunk eddy viscosity model is implemented in an RANS simulator and calibrated, using vorticity measurements, to infer three parameters. The calibration is Bayesian and is solved using a Markov chain Monte Carlo (MCMC) method. A 3D probability density distribution for the inferred parameters is constructed, thus quantifying the uncertainty in the estimate. The phenomenal cost of using a 3D flow simulator inside an MCMC loop is mitigated by using surrogate models ("curve-fits"). A support vector machine classifier (SVMC) is used to impose our prior belief regarding parameter values, specifically to exclude nonphysical parameter combinations. The calibrated model is compared, in terms of its predictive skill, to simulations using uncalibrated linear and CEVMs. We find that the calibrated model, with one quadratic term, is more accurate than the uncalibrated simulator. The model is also checked at a flow condition at which the model was not calibrated.
Fluctuating boundary layer pressure fluctuations are an important loading component for high speed reentry vehicles. Characterization of the unsteady time series requires access to longitudinal and lateral coherence expressions as well spatial correlation and frequency power-spectral density models. Coherence, spatial correlation and frequency power spectral density are related as through their cross-spectral density definitions. However the frequency PSD and the spatial correlation are often based upon measurements or approximate models which may introduce bias in the associated derived coherence function. Here, we examine the effect of measurement and model form associated with frequency spectrum and correlation on the longitudinal and lateral coherence for supersonic pressure fluctuation flow fields. The widely utilized Corcos separable coherence model functional form has been employed in this study. The associated integral equations which relate coherence and correlation are solved using a simple iterative approach. To minimize distortion in results due to computational issues a high accuracy numerical integration procedure is utilized. Despite a more robust computational approach, solution accuracy is limited for some problems by the functional form of the longitudinal coherence model. These limitations are discussed in detail. This overall approach is applied to Mach 5 and Mach 8 seven degree sharp cone pressure fluctuation measurements. Estimates for the parameters associated with the Corcos coherence expressions are typically larger than more traditional values especially for the longitudinal coherence. These larger values suggest that fluctuations streamwise correlation length is small. Limited longitudinal correlation can be associated with shock influence and is explored as a possible cause.
Wave packet analysis provides a connection between linear small disturbance theory and subsequent nonlinear turbulent spot flow behavior. The traditional association between linear stability analysis and nonlinear wave form is developed via the method of stationary phase whereby asymptotic (simplified) mean flow solutions are used to estimate dispersion behavior and stationary phase approximation are used to invert the associated Fourier transform. The resulting process typically requires nonlinear algebraic equations inversions that can be best performed numerically, which partially mitigates the value of the approximation as compared to a more complete, e.g. DNS or linear/nonlinear adjoint methods. To obtain a simpler, closed-form analytical result, the complete packet solution is modeled via approximate amplitude (linear convected kinematic wave initial value problem) and local sinusoidal (wave equation) expressions. Significantly, the initial value for the kinematic wave transport expression follows from a separable variable coefficient approximation to the linearized pressure fluctuation Poisson expression. The resulting amplitude solution, while approximate in nature, nonetheless, appears to mimic many of the global features, e.g. transitional flow intermittency and pressure fluctuation magnitude behavior. A low wave number wave packet models also recover meaningful auto-correlation and low frequency spectral behaviors.