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Exploiting Active Subspaces to Quantify Uncertainty in the Numerical Simulation of the HyShot II Scramjet
Paul Constantine, Michael Emory, Johan Larsson, Gianluca Iaccarino
TL;DR
The paper addresses uncertainty in hypersonic scramjet performance where expensive multiphysics simulations make conventional sampling impractical. It uses active subspaces to reduce seven operating parameters to one derived active variable. With 68 simulations across two fuel-injection conditions, the method supports bounds, safety classification, sensitivity analysis, and cumulative-distribution estimation.
Problem
Uncertainty-aware estimation of safe HyShot II operating limits is difficult because multiphysics simulations are expensive and prior studies largely ignored operating-condition uncertainty.
Method
Active subspaces use a global least-squares-fit linear approximation to reparameterize seven inputs through a one-dimensional active variable and validated univariate response surface.
Results
68 simulations across two fuel-pressure cases support active-subspace identification and uncertainty analyses including exit-pressure bounds, safe operating conditions, sensitivity, and a cumulative distribution function.
Takeaways & Limitations
A validated one-dimensional representation makes otherwise infeasible uncertainty quantification practical for the computationally expensive HyShot II model.
Abstract
from arXiv · showhide
We present a computational analysis of the reactive flow in a hypersonic scramjet engine with focus on effects of uncertainties in the operating conditions. We employ a novel methodology based on active subspaces to characterize the effects of the input uncertainty on the scramjet performance. The active subspace identifies one-dimensional structure in the map from simulation inputs to quantity of interest that allows us to reparameterize the operating conditions; instead of seven physical parameters, we can use a single derived active variable. This dimension reduction enables otherwise infeasible uncertainty quantification, considering the simulation cost of roughly 9500 CPU-hours per run. For two values of the fuel injection rate, we use a total of 68 simulations to (i) identify the parameters that contribute the most to the variation in the output quantity of interest, (ii) estimate upper and lower bounds on the quantity of interest, (iii) classify sets of operating conditions as safe or unsafe corresponding to a threshold on the output quantity of interest, and (iv) estimate a cumulative distribution function for the quantity of interest.
1. Introduction
HyShot II simulations address difficult, expensive multiphysics flow predictions and uncertain safe-operability limits. Active subspaces reduce seven uncertain inputs to a one-dimensional representation that supports uncertainty quantification and safety analysis.
- Motivation: Accurately predicting HyShot II internal flow is challenging because turbulence, shocks, boundary layers, mixing, and combustion interact across disparate scales.These modeling requirements make the simulations fairly expensive.
- Motivation: Maximum thrust occurs close to boundaries between supersonic, shock-train, and unstart regimes, making safe operation difficult to determine.Unstart is associated with a potentially catastrophic loss in thrust.
- Research gap: Previous studies largely ignored operating-condition uncertainty, leaving insufficient confidence in quantitative safe fuel-flow limits and encouraging experience-based safety margins.The paper motivates explicitly identifying, estimating, and accounting for these uncertainties.
- Contribution: Active subspaces represent the quantity of interest as a univariate function of a derived active variable instead of seven independent physical parameters.The approach uses a global least-squares-fit linear approximation and scales linearly with input dimension for this model.
- Contribution: The reduced representation supports cumulative-distribution estimation, quantity-of-interest bounds, safe-parameter identification, and sensitivity analysis.These analyses would otherwise be difficult because each simulation costs roughly 9500 CPU-hours.
2. Methodology for HyShot II simulations
The HyShot II model separates a cheaper 2D forebody and isolator from a fully 3D combustion chamber and nozzle. RANS is selected as the practical turbulence-treatment framework for uncertainty quantification, with fuel injection and combustion represented in the 3D domain.
- Geometry and computational grids: The computational model separates the intake ramp and isolator into 2D from the combustion chamber and exit nozzle into 3D to reduce simulation cost.The decomposition assumes spanwise-uniform combustion-chamber inflow upstream of fuel injection.
- Geometry and computational grids: The 2D domain uses a forebody/ramp grid with just under 50k control volumes, while the 3D domain uses a 1/8-span representation with 1.2M control volumes.The reduced 3D span exploits symmetry and includes the injection-port and nozzle geometry.
- Domain coupling: The 3D oxidizer inflow is a wall-normal profile extracted from the 2D simulation at x = 352.68mm and applied uniformly across the combustor inlet.The extraction location captures the body-wall oblique shock above the boundary-layer-resolving cells.
- Reynolds-averaged Navier-Stokes: RANS is used for uncertainty quantification because LES is too costly for the many simulations needed to estimate solution statistics.RANS reduces cost by modeling time-averaged flow with a coarser mesh, while introducing dependence on turbulence-model accuracy.
- Reynolds-averaged Navier-Stokes: The simulations use an SST-based turbulence closure with limiter modifications and model pressure predictions that previously showed agreement with experimental combustion-chamber compression ratios.The SST formulation is widely used for compressible boundary layers and shock-turbulence interactions.
Turbulence.
The turbulence treatment uses limiter modifications to the SST-based RANS closure. These modifications target eddy viscosity and turbulence-kinetic-energy production to maintain realizable modeled stresses.
- Turbulence: The model applies an eddy-viscosity limiter and a turbulence-kinetic-energy production limiter to the SST closure.The production limiter is intended to ensure realizable Reynolds stresses.
Transition.
Boundary-layer transition is represented by manually specified transition locations on the intake ramp, body wall, and cowl wall. These locations modify turbulence-production and destruction behavior upstream.
- Transition: Manual transition locations on the intake ramp, body wall, and cowl wall mimic the γ−Reθt transition model through inhibited turbulence production and destruction.The prescribed locations affect boundary-layer flow upstream of each transition point.
Combustion.
The combustion model uses a flamelet/progress variable approach that tabulates laminar flamelet chemistry and accounts for turbulence through a presumed beta distribution of mixture fraction.
- The FPVA model tabulates chemistry as laminar flamelet solutions indexed by boundary conditions and background pressure.
- Turbulence effects on combustion are approximated with a presumed beta probability density for the fuel/air mixture fraction.
Boundary conditions.
The simulations solve steady compressible flow, turbulence, and combustion equations with specified wall, outlet, freestream, and fuel-inlet conditions.
- Walls are isothermal at Tw = 300K, while the nozzle, bleed channel, and freestream use Neumann boundary conditions.
- Fuel-inlet conditions specify stagnation pressure P0,H2 and stagnation temperature T0,H2, with T0,H2 fixed at 300K.
- The model solves steady compressible RANS, k−ω SST turbulence, and FPVA combustion equations coupled to a lookup table.
Cost.
The uncertainty-quantification simulations are computationally expensive, especially in three dimensions.
- Each 3D combustor simulation averaged approximately 76 hours on 120 CPUs, with substantial variation in iterations required for convergence.
- Each 2D forebody simulation averaged approximately 8.1 hours on 48 CPUs.
3. Uncertainty sources and the quantity of interest
The study characterizes uncertain operating conditions and propagates them through the HyShot II model to quantify uncertainty in integrated exit pressure. It uses measured and expert-informed ranges, with the pressure integral treated as the performance quantity of interest.
- 3. Uncertainty sources and the quantity of interest: Uncertainty quantification uses input ranges informed by observations, theory, and expert opinion to estimate output bounds and safe operating sets.
- 3. Uncertainty sources and the quantity of interest: A uniform input density is selected by Jaynes’ maximum entropy principle when no additional measurement information is available.
- 3.1. Inflow conditions, mean quantities: Fuel plenum pressure is used as the control parameter because it was measured directly, although equivalence ratio has a direct chemical interpretation.
- 3.3. Transition locations: The 2D uncertainty analysis considers transition locations xt,r and xt,c because the corresponding simulations are substantially cheaper than 3D computations.
- 3.3. Transition locations: Transition uncertainty is represented through the critical value Reθ/Me = 200 rather than directly assigning uncertainty to xt,r.
- 3. Uncertainty sources and the quantity of interest: The integrated exit pressure over 0.64 ≤ x ≤ 0.65m is the quantity of interest used to assess scramjet performance.
4. Uncertainty quantification methodology
The methodology reduces uncertainty quantification for an expensive seven-parameter scramjet simulation by identifying low-dimensional structure in its quantity of interest. It uses active-subspace and regression-based dimension reduction to estimate bounds, safe-operation regions, and the output distribution despite limited evaluations.
- Targets: The uncertainty quantification targets bounds on f(x), inputs yielding safe operation below a threshold, and the cumulative distribution function of f(x).Here f(x) denotes integrated exit pressure as a function of seven normalized input parameters.
- Computational challenge: Each quantity-of-interest evaluation costs approximately 9500 CPU-hours, making exhaustive sampling of the seven-dimensional input space infeasible.The output also lacks exploitable prior structure, gradients, and Hessians, while numerical error further complicates uncertainty analysis.
- Active-subspace reduction: Active subspaces identify important input directions from eigenvectors of a gradient-based matrix, enabling low-dimensional approximations of f(x).Eigenvalues measure average changes in f along corresponding eigenvectors, and a spectral gap supports retaining the leading directions.
- Active-subspace estimation: Because gradients are unavailable and finite differences are unreliable, the method estimates the active direction using a least-squares linear approximation from sampled simulations.The procedure draws samples, evaluates f, fits linear coefficients, and normalizes the coefficient vector to obtain w.
- Validation: A summary plot of w^T x versus f(x) tests whether the output is approximately a univariate function of the active variable.Departures can reflect variation orthogonal to w or too few samples, while bootstrap estimates variability in the computed direction.
5. Discovering and exploiting a one-dimensional active subspace in the HyShot II quantity of interest
The study finds that HyShot II exit pressure can be represented accurately with a one-dimensional active variable derived from seven uncertain inputs. This reduction supports pressure-range estimation, safe-operation classification, and uncertainty quantification at much lower cost than exhaustive simulation.
- 5. Discovering and exploiting a one-dimensional active subspace in the HyShot II quantity of interest: The seven-input exit-pressure map exhibits a sufficiently accurate univariate relationship with w^T x at both fuel-plenum pressures, enabling the three uncertainty-quantification computations.The active-variable summary plots support this approximation for both 4.8 bar and 5.6 bar.
- 5.1. Sensitivity analysis: Four inputs dominate the active subspace, with angle of attack, stagnation temperature and pressure, and turbulence intensity contributing most to exit-pressure variation.The interpretation is qualified because SST-type RANS models can behave erratically across strong shock waves.
- 5.1. Sensitivity analysis: The leading sensitivity factors remain similar between 4.8 and 5.6 bar, but stagnation pressure and enthalpy become relatively more important near the regime boundary.The paper attributes this plausibly to changes in wall heat losses and frictional momentum losses, while total heat release remains essentially set by fuel mass flux.
- 5.3. Approximating the range of exit pressures: The estimated exit-pressure range requires only four additional simulations and is supported by the observed monotonic active-variable structure.The endpoint heuristic may bound the initial samples or estimate the full input range, but the authors characterize it as informed guess-and-check.
- 5. Discovering and exploiting a one-dimensional active subspace in the HyShot II quantity of interest: R^2 = 0.993 at 4.8 bar and R^2 = 0.998 at 5.6 bar indicate close quadratic fits for exit pressure as a function of the active variable.The authors treat these values as measures of discrepancy rather than statistical goodness of fit because the simulations contain no random noise.
- 5.4. Constraining the exit pressure: The active-subspace representation converts a pressure threshold into a safe input region whose parameter relationships can be physically interpreted after shifting and scaling.For example, the safe angle-of-attack range depends on the other input variables; Table 4 reports the affected parameter ranges.
6. Summary and discussion
The study uses active subspaces to reduce seven uncertain scramjet inputs to a one-dimensional representation of exit pressure, enabling uncertainty quantification with 68 full simulations. This approach estimates pressure ranges, safe operating conditions, and exit-pressure distributions while identifying influential parameters.
- A near one-to-one relationship between the active variable and exit pressure makes the seven-input uncertainty problem computationally tractable.The smaller-pressure case uses 50 simulations, while the larger-pressure case uses 14, whose bootstrap estimate is more variable.
- Active-subspace coefficients serve as sensitivity metrics, revealing which input perturbations produce the greatest changes in exit pressure.This provides physical insight alongside the uncertainty estimates.
- 68 full simulations quantify exit-pressure uncertainty for two fuel plenum pressures using one-dimensional active-subspace models.The simulations include 64 runs to identify active subspaces and 4 to estimate exit-pressure ranges.
- The validated monotonic response surface estimates exit-pressure bounds, classifies safe operating conditions, and approximates the cumulative distribution function.For a continuous monotonic function, extrema occur at the endpoints of the active-variable interval.
- The approach is limited to cases where summary plots support a univariate, monotonic relationship between the active variable and the quantity of interest.Without that structure, the range and response-surface validation heuristics would not be justified.