Efficient Uncertainty Quantification & Sensitivity Analysis for Hypersonic Flow and Material Response Simulations Under Inherent and Model-Form Uncertainties

Accurate numerical prediction of coupled hypersonic flow fields and ablative TPS material response is challenging due to the complex nature of the physics. The uncertainties associated with various physical models used in high-enthalpy hypersonic flow and material response simulations can have significant effects on the accuracy of the results including the heat-flux and temperature distributions in various layers of ablating TPS material. These uncertainties can arise from the lack of knowledge in physical modeling (model-form or epistemic uncertainty) or inherent variations in the model inputs (aleatory or probabilistic uncertainty). It is important to include both types of uncertainty in the simulations to properly assess the accuracy of the results and to design robust and reliable TPS for reentry or hypersonic cruise vehicles. In addition to the quantification of uncertainties, global sensitivity information for the output quantities of interest play an important role for the ranking of the contribution of each uncertainty source to the overall uncertainty, which may be used for the proper allocation of resources in the improvement of the physical models or reduce the number of uncertain variables to be considered in the uncertainty analysis. The uncertainty quantification for coupled high-fidelity hypersonic flow and material response predictions can be challenging due to the computational expense of the simulations, existence of both model-form and inherent uncertainty sources, large number of uncertain variables, and highly non-linear relations between the uncertain variables and the output response variables. The objective of this talk will be to introduce a computationally efficient and accurate uncertainty quantification (UQ) and global sensitivity analysis approach for potential application to coupled aerothermodynamics and material response simulations, which is being developed to address the aforementioned challenges. The UQ approach to be described is based on the second-order uncertainty quantification theory utilizing a stochastic response surface obtained with non-intrusive polynomial chaos and is capable of efficiently propagating both the inherent and the model-form uncertainties in the physical models. The non-intrusive nature of the UQ approach requires no modification to the deterministic codes, which is a significant benefit for the complex numerical simulation considered in this problem. The global non-linear sensitivity analysis to be introduced is based on variance decomposition, which again utilizes the polynomial chaos expansions. In addition to the description of the UQ approach, the talk will also include the presentation of UQ results from a recent demonstration of the methodology, which included the uncertainty quantification and sensitivity analysis of surface heat-flux on the spherical heat shield of a reentry vehicle (a case selected from CUBRC experimental database). This study involved the use of NASA DPLR code and the treatment of the free-stream velocity (inherent uncertainty), collision integrals for the transport coefficients (model-form uncertainty), and the surface catalysis (model-form uncertainty) as uncertain variables. The talk will also include the description of an adaptive UQ framework being developed as part of a NASA JPL STTR project to quantify the uncertainty in multi-physics spacecraft simulations with large number of uncertain variables

Inherent and Model-Form Uncertainty Analysis for CFD Simulation of Synthetic Jet Actuators

A mixed (aleatory and epistemic) uncertainty quantification (UQ) method was applied to computational uid dynamics (CFD) modeling of a synthetic jet actuator. A test case, (ow over a hump model with synthetic jet actuator control) from the CFDVAL2004 work-shop was selected to apply the Second-Order Probability framework implemented with a stochastic response surface obtained from Quadrature-Based Non-Intrusive Polynomial Chaos (NIPC). Three uncertainty sources were considered: (1) epistemic (model-form) uncertainty in turbulence model, (2) aleatory (inherent) uncertainty in free stream veloc-ity and (3) aleatory uncertainty in actuation frequency. Uncertainties in both long-time averaged and phase averaged quantities were quantified using a fourth order polynomial chaos expansion (PCE). A global sensitivity analysis with Sobol indices was utilized to rank the importance of each uncertainty source to the overall output uncertainty. The results indicated that for the long-time averaged separation bubble size, the uncertainty in turbulence model had a dominant contribution, which was also observed in the long-time averaged skin friction coeficients at three selected locations. The mixed uncertainty results for phase averaged x-velocity distributions at three selected locations showed that the 95% confidence interval (CI) could generally envelope the experimental data. The Sobol indices showed that near the wall, the uncertainty in turbulence model had a main inuence on the x-velocity, while approaching the main stream, the uncertainty in free stream velocity be-came a larger contributor. The mixed uncertainty quantification approach demonstrated in this study can also be applied to other computational uid dynamics problems with inherent and model-form input uncertainities

Application of a Six Degrees-of-Freedom Drag Model for Small Satellite Mission Development

For spacecraft in low-perigee orbits, atmospheric drag presents one of the largest uncertainties in dynamics modeling. These uncertainties are particularly relevant to small satellites, which often fly in the LEO regime and produce control forces and torques comparable in magnitude to drag. In this study, a six degrees-of-freedom orbital dynamics model with drag perturbations is developed, and several applications of the model are investigated. The model is used to evaluate differential drag dynamics for the MR and MRS SAT microsatellite pair, and the implications to collision avoidance and end-of-life procedures are discussed. Preliminary propellant usage estimates for the mission are also generated. A modified method for determining ballistic coefficient using relative satellite navigation data is introduced and compared to previous methods

Efficient Uncertainty Quantification Applied to the Aeroelastic Analysis of a Transonic Wing

The application of a Point-Collocation Non-Intrusive Polynomial Chaos method to the uncertainty quantification of a stochastic transonic aeroelastic wing problem has been demonstrated. The variation in the transient response of the first aeroelastic mode of a three-dimensional wing in transonic flow due to the uncertainty in free-stream Mach number and angle of attack was studied. A curve-fitting procedure was used to obtain time-independent parameterization of the transient aeroelastic responses. Among the uncertain parameters that characterize the time-dependent transients, the damping factor was chosen for uncertainty quantification, since this parameter can be thought as an indicator for flutter. Along with the mean and the standard deviation of the damping factor, the probability of having flutter for the given uncertainty in the Mach number and the angle of attack has been also calculated. Besides the Point-Collocation Non-Intrusive Polynomial Chaos method, 1000 Latin Hypercube Monte Carlo simulations were also performed to quantify the uncertainty in the damping factor. The results obtained for various statistics of the damping factor including the flutter probability showed that an 8th degree Point-Collocation Non-Intrusive Polynomial Chaos expansion is capable of estimating the statistics at an accuracy level of 1000 Latin Hypercube Monte Carlo simulation with a significantly lower computational cost. In addition to the uncertainty quantification, the response surface approximation, sensitivity analysis, and reconstruction of the transient response via Non-Intrusive Polynomial Chaos were also demonstrated

Uncertainty Quantification Integrated to the CFD Modeling of Synthetic Jet Actuators

The Point Collocation Non-Intrusive Polynomial Chaos (NIPC) method has been applied to two stochastic synthetic jet actuator problems used as test cases in the CFDVAL2004 workshop to demonstrate the integration of computationally efficient uncertainty quantification to the high-fidelity CFD modeling of synthetic jet actuators. In Case1 where the synthetic jet is issued into quiescent air, the NIPC method is used to quantify the uncertainty in the long-time averaged u and v-velocities at several locations in the flow field, due to the uniformly distributed uncertainty introduced in the amplitude and frequency of the oscillation of the piezo-electric membrane. Fifth order NIPC expansions were used to obtain the uncertainty information, which showed that the variation in the v-velocity is high in the region directly above the jet slot and the variation in the u-velocity is maximum in the region immediately adjacent to the slot. Even with a ten percent variation in the amplitude and frequency, the long-time averaged u and v velocity profiles could not match the experimental measurements at y=0.1mm above the slot indicating that the discrepancy may be due to other uncertainty sources in CFD or measurement errors. In Case 2 which includes a cross flow, the free stream velocity is treated as an uncertain input variable. Fifth degree NIPC expansions were employed to quantify the uncertainty in phase averaged velocity profiles as well as long-time averaged wall pressure and skin friction coefficient distributions. The results of Case 2 show that the uncertainty in phase averaged velocity profiles gets larger when approaching the main stream. The size of a separation bubble observed in this case remains relatively insensitive to the uncertain free stream velocity within the tolerance range considered

Uncertainty Quantification Integrated to CFD Modeling of Synthetic Jet Actuators

The Point-Collocation Non-intrusive Polynomial Chaos (NIPC) method has been applied to a stochastic synthetic jet actuator problem used as one of the test cases in the CFDVAL2004 workshop to demonstrate the integration of computationally efficient uncertainty quantification to the high-fidelity CFD modeling of synthetic jet actuators. The test case included the simulation of an actuator generating a synthetic jet issued into quiescent air. The Point-Collocation NIPC method is used to quantify the uncertainty in the long-time averaged u and v-velocities at several locations in the flow field due to the uniformly distributed uncertainty introduced in the amplitude and frequency of the oscillation of the piezo-electric membrane. Fifth-order NIPC expansions were used to obtain the uncertainty information, which showed that the variation in the v-velocity is high in the region directly above the jet slot and the variation in the u-velocity is maximum in the region immediately adjacent to the slot. Even with a ±5% variation in the amplitude and frequency, the long-time averaged u and v-velocity profiles could not match the experimental measurements at y=0.1 mm above the slot indicating that the discrepancy may be due to other uncertainty sources in CFD and/or due to the measurement errors

Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables

The accuracy and the computational efficiency of a Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) method applied to stochastic problems with multiple uncertain input variables has been investigated. Two stochastic model problems with multiple uniform random variables were studied to determine the effect of different sampling methods (Random, Latin Hypercube, and Hammersley) for the selection of the collocation points. The effect of the number of collocation points on the accuracy of polynomial chaos expansions were also investigated. The results of the stochastic model problems show that all three sampling methods exhibit a similar performance in terms of the the accuracy and the computational efficiency of the chaos expansions. It has been observed that using a number of collocation points that is twice more than the minimum number required gives a better approximation to the statistics at each polynomial degree. This improvement can be related to the increase of the accuracy of the polynomial coefficients due to the use of more information in their calculation. The results of the stochastic model problems also indicate that for problems with multiple random variables, improving the accuracy of polynomial chaos coefficients in NIPC approaches may reduce the computational expense by achieving the same accuracy level with a lower order polynomial expansion. To demonstrate the application of Point-Collocation NIPC to an aerospace problem with multiple uncertain input variables, a stochastic computational aerodynamics problem which includes the numerical simulation of steady, inviscid, transonic flow over a three-dimensional wing with an uncertain free-stream Mach number and angle of attack has been studied. For this study, a 5th degree Point-Collocation NIPC expansion obtained with Hammersley sampling was capable of estimating the statistics at an accuracy level of 1000 Latin Hypercube Monte Carlo simulations with a significantly lower computational cost

A Non-Intrusive Polynomial Chaos Method for Uncertainty Propagation in CFD Simulations

An inexpensive non-intrusive polynomial chaos (NIPC) method for the propagation of input uncertainty in CFD simulations is presented. The method is straightforward to implement for any stochastic fluid dynamics problem and computationally less expensive than sampling or quadrature based non-intrusive methods. To validate the present NIPC approach, the method has been applied to: (1) an inviscid oblique shock wave problem with geometric uncertainty, (2) an inviscid expansion wave problem with geometric uncertainty, and (3) a subsonic, two-dimensional, laminar boundary layer flow over a flat plate with an uncertain free-stream dynamic viscosity. For all test cases, the statistics (mean and the standard deviation) obtained with the NIPC method were in good agreement with the results of the Monte Carlo simulations. A fourth order polynomial chaos expansion was sufficient to approximate the statistics and the shape of the output uncertainty distributions with the desired accuracy. Only in the shock region of the first test case a sixth order polynomial expansion was required to estimate the statistics of pressure within the 95% confidence intervals of the Monte Carlo results, since the shape of the distributions obtained with 3rd order spatially accurate Euler solutions were highly non-Gaussian in this region

Analysis of Jet-Wing Distributed Propulsion from Thick Wing Trailing Edges

Conventional airliners use two to four engines in a Cayley-type arrangement to provide thrust, and the thrust is concentrated right behind the engine. Distributed propulsion is the idea of redistributing the thrust across most, or all, of the wingspan of an aircraft. This can be accomplished by using several large engines and using a duct to spread out the exhaust flow to form a jet-wing or by using many small engines spaced along the span of the wing. Jet-wing distributed propulsion was originally suggested as a way to improve propulsive efficiency. A previous study at Virginia Tech assessed the potential gains in propulsive efficiency. The purpose of this study was to assess the performance benefits of jet-wing distributed propulsion. The Reynolds-averaged, finitevolume, Navier-Stokes code GASP was used to perform parametric computational fluid dynamics (CFD) analyses on two-dimensional jet-wing models. The jet-wing was modeled by applying jet boundary conditions on the trailing edges of blunt trailing edge airfoils such that the vehicle was self-propelled. As this work was part of a Blended-Wing-Body (BWB) distributed propulsion multidisciplinary optimization (MDO) study, two airfoils of different thickness were modeled at BWB cruise conditions as examples. One airfoil, representative of an outboard BWB wing section, was 11% thick. The other airfoil, representative of an inboard BWB wing section, was 18% thick. Furthermore, in an attempt to increase the propulsive efficiency, the trailing edge thickness of the 11% thick airfoil was doubled in size. The studies show that jet-wing distributed propulsion can be used to obtain propulsive efficiencies on the order of turbofan engine aircraft. If the trailing edge thickness is expanded, then jet-wing distributed propulsion can give improved propulsive efficiency. However, expanding the trailing edge must be done with care, as there is a drag penalty