The goal of this research was to develop a source term module to model the effect of cavity unsteadiness in gas-path only simulations.In this approach the unsteady effect of the cavity on the main gas-path solution is modeled by adding source terms to the right hand side of the Navier-Stokes equations instead of resolving the cavity geometry. This idea increases considerably the numerical efficiency of the scheme by avoiding the computation of small-scale fluid dynamic structures and the geometric details of the cavity.The paper is a two-part work. In the first part the source terms are developed via a lumped deterministic source term (LDST) approach using unsteady complete cavity geometry simulations. A simplified rectangular cavity has been used for the present work. A parametric study of the source term variation with Mach number is presented. Steady simulations of the passage only (i.e., sans cavity) with source terms inserted have been performed and the results obtained compared to time-averaged solutions that resolve the cavity. The steady state solution using source terms was able to correctly incorporate all unsteady effects present in the time average of the unsteady cavity solution.The second part of the paper discusses the modeling of cavity source terms via back-propagation neural networks. Sample analyses for subsonic Mach numbers, input pressures and boundary layer thicknesses, are presented. A background neural network is trained to generate a source term for any combination of parameters. The resultant flow field is computed for a finite number of test cases. Neural networks are utilized to model the complex data dependencies instead of computing source terms for every test case. The neural network can be trained to generate quickly a source term for any combination of input parameters. The steady state flow through the passage including the unsteady effect of the cavity was obtained using neural network generated source terms. The resultant flow fields are only slightly different from the calculated time average results and thus encourage the further development of this method.

The goal of this research was to develop a source term module to model the effect of cavity unsteadiness in gas-path only simulations. In this approach the unsteady effect of the cavity on the main gas-path solution is modeled by adding source terms to the right hand side of the Navier-Stokes equations instead of resolving the cavity geometry. This idea increases considerably the numerical efficiency of the scheme by avoiding the computation of small-scale fluid dynamic structures and the geometric details of the cavity. The paper is a two-part work. In the first part the source terms are developed via a lumped deterministic source term (LDST) approach using unsteady complete cavity geometry simulations. A simplified rectangular cavity has been used for the present work. A parametric study of the source term variation with Mach number is presented. Steady simulations of the passage only (i.e., sans cavity) with source terms inserted have been performed and the results obtained compared to timeaveraged solutions that resolve the cavity. The steady state solution using source terms was able to correctly incorporate all unsteady effects present in the time average of the unsteady cavity solution. The second part of the paper discusses the modeling of cavity source terms via back-propagation neural networks. Sample analyses for subsonic Mach numbers, input pressures and boundary layer thicknesses, are presented. A background neural network is trained to generate a source term for any combination of parameters. The resultant flow field is computed for a finite number of test cases. Neural networks are then utilized to model the complex data dependencies instead of computing source terms for every test case. The neural network can be trained to generate quickly a source term for any combination of input parameters. The steady state flow through the passage including the unsteady effect of the cavity was obtained using neural network generated source terms. The resultant flow fields are only slightly different from the calculated time average results and thus encourage the further development of this method.

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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

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