Bayesian Mass Averaging in Rigs and Engines

This paper introduces the Bayesian mass average and details its computation. Owing to the complexity of flow in an engine and the limited instrumentation and the precision of the sensor apparatus used, it is difficult to rigorously calculate mass averages. Building upon related work, this paper views any thermodynamic quantity's spatial variation at an axial plane in an engine (or a rig) as a Gaussian random field. In cases where the mass flow rate is constant in the circumferential direction but can be expressed via a polynomial or spline radially, this paper presents an analytical calculation of the Bayesian mass average. In cases where the mass flow rate itself can be expressed as a Gaussian random field, a sampling procedure is presented to calculate the Bayesian mass average. Examples of the calculation of the Bayesian mass average for temperature are presented, including with a real engine case study where velocity profiles are inferred from stagnation pressure measurements.

A linear stochastic biharmonic heat equation: hitting probabilities

Consider the linear stochastic biharmonic heat equation on a d–dimen- sional torus ($$d=1,2,3$$ d = 1 , 2 , 3 ), driven by a space-time white noise and with periodic boundary conditions: $$\begin{aligned} \left( \frac{\partial }{\partial t}+(-\varDelta )^2\right) v(t,x)= \sigma \dot{W}(t,x),\ (t,x)\in (0,T]\times {\mathbb {T}}^d, \end{aligned}$$ ∂ ∂ t + ( - Δ ) 2 v ( t , x ) = σ W ˙ ( t , x ) , ( t , x ) ∈ ( 0 , T ] × T d , $$v(0,x)=v_0(x)$$ v ( 0 , x ) = v 0 ( x ) . We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for $$d=2$$ d = 2 , they include a $$z(\log \tfrac{c}{z})^{1/2}$$ z ( log c z ) 1 / 2 term. Consider D independent copies of the random field solution to (0.1). Applying the criteria proved in Hinojosa-Calleja and Sanz-Solé (Stoch PDE Anal Comp 2021. 10.1007/s40072-021-00190-1), we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets.This yields results on the polarity of sets and on the Hausdorff dimension of the path process.