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.
[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.
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. In this paper, 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. Finally, 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.
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.
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.
Reynolds-averaged Navier–Stokes models are not very accurate for high-Reynolds-number compressible jet-in-crossflow interactions. The inaccuracy arises from the use of inappropriate model parameters and model-form errors in the Reynolds-averaged Navier–Stokes model. In this study, 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.
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.
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
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.
We demonstrate a Bayesian method that can be used to calibrate computationally expensive 3D RANS (Reynolds Av- eraged Navier Stokes) 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 parameter estimates. Methods such as Markov chain Monte Carlo (MCMC) estimate the PDF by sampling, with each sample requiring a run of the RANS model. Consequently a quick-running surrogate is used instead to 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 parameter being estimated is complex. We show how the training data used to construct the surrogate can 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 μ, C ε2 , C ε1 ) 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 cal- ibration data. We also check the limit of applicability of the calibration by testing at off-calibration flow regimes. We find that calibration yield turbulence model parameters which predict the flowfield far better than when the nomi- nal values of the parameters are used. Substantial improvements are still obtained when we use the calibrated RANS model to predict jet-in-crossflow at Mach numbers and jet strengths quite different from those used to generate the ex- perimental (calibration) data. Thus the primary reason for poor predictive skill of RANS, when using nominal values of the turbulence model parameters, was parametric uncertainty, which was rectified by calibration. Post-calibration, the dominant contribution to model inaccuraries are due to the structural errors in RANS.
Here we discuss an improved Corcos (Corcos (1963), (1963)) style cross spectral density utilizing zero pressure gradient, supersonic (Beresh et. al. (2013)) data sets. Using the connection between narrow band measurements with broadband cross-spectral density, i.e. Γ(ξ ,η ,ω )= Φ (ω) A(ωη/U )exp (-i ωξ/U) we focus on estimating coherence expressions of the form: A (ξω nb/U) and B (ηω nb/ U) where ωnb denotes the narrow band frequency, i.e. the band center frequency value and ξ and η are sensors spacing in streamwise/longitudinal and cross-stream/lateral directions, respectively. A methodology to estimate the parameters which retains the Corcos exponential functional form, A(ξω/U)=exp(-klat ηω/U) but identifies new parameters (constants) consistent with the Beresh et. al. data sets is discussed. The Corcos result requires that the data be properly explained by self-similar variable: ξω/U and ηω/U. The longitudinal (streamwise) variable ξω/U tends to provide a better data collapse, while, consistent with the literature the lateral ηω/U is only successful for higher band center frequencies. Assuming the similarity variables provide a useful description of the data, the longitudinal coherence decay constant result using the Beresh et. al. data sets yields a value for the longitudinal constant klong≈0.36-0.28 that is approximately 3x larger than the “traditional” (low speed, large Reynolds number and zero pressure gradient) of klong≈0.11. We suggest that the most likely reason that the Beresh et. al. data sets incur increased longitudinal decay which results in reduced coherence lengths is due to wall shear induced compression causing an adverse pressure gradient. Focusing on the higher band center frequency measurements where the frequency dependent similarity variables are applicable, the lateral or transverse coherence decay constant klat≈0.7 is consistent with the “traditional” (low speed, large Reynolds number and zero pressure gradient). It should be noted, that the longitudinal/streamwise coherence decay deviates from the value observed by other researchers while the lateral/ cross-stream value is consistent has been observed by other researchers. We believe that while the measurements used to obtain new decay constant estimates are from internal wind tunnel tests, they likely provide a useful estimate expected reentry flow behavior and are therefore recommended for use. These data could also be useful in determining the uncertainty of correlation length for a uncertainty quantification (UQ) analysis.
We develop a novel calibration approach to address the problem of predictive ke RANS simulations of jet-incrossflow. Our approach is based on the hypothesis that predictive ke parameters can be obtained by estimating them from a strongly vortical flow, specifically, flow over a square cylinder. In this study, we estimate three ke parameters, C%CE%BC, Ce2 and Ce1 by fitting 2D RANS simulations to experimental data. We use polynomial surrogates of 2D RANS for this purpose. We conduct an ensemble of 2D RANS runs using samples of (C%CE%BC;Ce2;Ce1) and regress Reynolds stresses to the samples using a simple polynomial. We then use this surrogate of the 2D RANS model to infer a joint distribution for the ke parameters by solving a Bayesian inverse problem, conditioned on the experimental data. The calibrated (C%CE%BC;Ce2;Ce1) distribution is used to seed an ensemble of 3D jet-in-crossflow simulations. We compare the ensemble's predictions of the flowfield, at two planes, to PIV measurements and estimate the predictive skill of the calibrated 3D RANS model. We also compare it against 3D RANS predictions using the nominal (uncalibrated) values of (C%CE%BC;Ce2;Ce1), and find that calibration delivers a significant improvement to the predictive skill of the 3D RANS model. We repeat the calibration using surrogate models based on kriging and find that the calibration, based on these more accurate models, is not much better that those obtained with simple polynomial surrogates. We discuss the reasons for this rather surprising outcome.
This report summarizes the work completed during FY2007 and FY2008 for the LDRD project ''Hybrid Plasma Modeling''. The goal of this project was to develop hybrid methods to model plasmas across the non-continuum-to-continuum collisionality spectrum. The primary methodology to span these regimes was to couple a kinetic method (e.g., Particle-In-Cell) in the non-continuum regions to a continuum PDE-based method (e.g., finite differences) in continuum regions. The interface between the two would be adjusted dynamically ased on statistical sampling of the kinetic results. Although originally a three-year project, it became clear during the second year (FY2008) that there were not sufficient resources to complete the project and it was terminated mid-year.
Class 8 tractor-trailers are responsible for 11-12% of the total US consumption of petroleum. Overcoming aero drag represents 65% of energy expenditure at highway speeds. Most of the drag results from pressure differences and reducing highway speeds is very effective. The goal is to reduce aerodynamic drag by 25% which would translate to 12% improved fuel economy or 4,200 million gal/year. Objectives are: (1) In support of DOE's mission, provide guidance to industry in the reduction of aerodynamic drag; (2) To shorten and improve design process, establish a database of experimental, computational, and conceptual design information; (3) Demonstrate new drag-reduction techniques; and (4) Get devices on the road. Some accomplishments are: (1) Concepts developed/tested that exceeded 25% drag reduction goal; (2) Insight and guidelines for drag reduction provided to industry through computations and experiments; (3) Joined with industry in getting devices on the road and providing design concepts through virtual modeling and testing; and (4) International recognition achieved through open documentation and database.
At 70 miles per hour, overcoming aerodynamic drag represents about 65% of the total energy expenditure for a typical heavy truck vehicle. The goal of this US Department of Energy supported consortium is to establish a clear understanding of the drag producing flow phenomena. This is being accomplished through joint experiments and computations, leading to the smart design of drag reducing devices. This paper will describe our objective and approach, provide an overview of our efforts and accomplishments, and discuss our future direction.
The Detached Eddy Simulation (DES) and steadystate Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling approaches are examined for the incompressible flow over a square cross-section cylinder at a Reynolds number of 21,400. A compressible flow code is used which employes a second-order Roe upwind spatial discretization. Efforts are made to assess the numerical accuracy of the DES predictions with regards to statistical convergence, iterative convergence, and temporal and spatial discretization error. Three-dimensional DES simulations compared well with two-dimensional DES simulations, suggesting that the dominant vortex shedding mechanism is effectively two-dimensional. The two-dimensional simulations are validated via comparison to experimental data for mean and RMS velocities as well as Reynolds stress in the cylinder wake. The steady-state RANS models significantly overpredict the size of the recirculation zone, thus underpredicting the drag coefficient relative to the experimental value. The DES model is found to give good agreement with the experimental velocity data in the wake, drag coefficient, and recirculation zone length.
Though not discussed extensively in the literature, it is known among workers in impact and penetration dynamics, e.g. the CTH analysis and development team at Sandia National Laboratories, that curvature of thin plates has a minimal effect on the penetration hole diameter due to a hypervelocity impact. To understand why curvature introduces a minimal effect on penetration hole size we extend a flat plate penetration hole diameter relationship to include the effect of body curvature. The effect of the body curvature on the hole diameter is shown to scale according to the dimensionless plate thickness to radius of curvature of the body i.e. h/R, which is typically small. Indeed for most problems where a single layer shell (plate) can be meaningfully defined, the effect of curvature upon hole diameter is on the order of other uncertainties in the problem, e.g. doubts concerning the appropriate equation of state and strength model, and is often, therefore, negligible.