Neural Network Modeling of Deterministic Unsteadiness Source Terms

Abstract

A neural-network-based lumped deterministic source term technique is presented that results in the prediction of an approximate time-average solution when used to modify a steady-state solver. Three different neural networks are developed for simple cavity flows using Mach number, cavity length-to-depth ratio, and aft wall translation as parameters. The results indicate that axial force data can be reproduced with less than 15% error as compared to the time average of a fully unsteady calculation. Computation times for the resultant neural network lumped deterministic source term approach were up to two orders of magnitude less than the comparable unsteady solution and were essentially identical to that of a steady-state calculation, although it must be noted that a database of unsteady calculations is required to develop the technique. The lumped deterministic source terms did not appear to affect the robustness of the steady-state solver adversely. NomenclatureA = control surface area a, a n i = neural network output neuron b l i = neural network bias c, c i = neural network input neuron D = cavity length E = error vector function e = neural network error vector e t = total energy F = force vector f = function k = turbulent kinetic energy L = cavity length L/D = cavity length-to-depth ratio M = Mach number n = normal direction vector n j i = neural network neuron P = primitive variable vector p= y-direction velocity .Sekar@wpafb.af.mi. Associate Fellow AIAA.[W]= weight factor matrix w = z-direction velocity (x, y, z) = Cartesian coordinate directions y + = nondimensional turbulent near-wall distance γ = ratio of specific heats ε = turbulent dissipation μ = molecular viscosity μ t = turbulent viscosity ρ = density υ = kinematic viscosity Subscripts wall = wall value 0 = stagnation conditions Superscripts -= time average = = derived quantity time average ∼ = deterministic fluctuation = stochastic fluctuation

Multiscale block spectral solution for unsteady flows

“…More closely relevant to the present interest in dealing with self‐excited short scale unsteadiness is the work of Ning and He, showing that by adding “unsteady stresses” as extra momentum and energy source terms, it would be possible to obtain the time‐averaged flow field with a purely steady flow solver for self‐excited vortex shedding around a cylinder or a blade trailing edge. This “unsteady stress” method has been so far however only applied to self‐excited unsteady flows for simple and isolated geometrical configurations (Ning and He and Lokovic et al).…”

Deterministic Stress Modeling for Multistage Compressor Flowfields

“…Finally, lumped deterministic stresses can be deduced from unsteady computations as proposed by Sondak et al [10]. Lukovic et al [11] recommended a neural network model for deterministic source terms, which would require a large database of unsteady calculations. In this paper two different modeling startegies are proposed.…”