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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Characteristics of Fluid‐Solid Interaction Constitutive Models Within Poroelastodynamics at Higher Strain‐Rates and Large Deformations Implemented in 1D

The large deformation, mixed formulation, finite element (FE) modeling approach presented in Irwin et al. 2024 is extended herein to include improved constitutive models for representing dynamic solid-fluid interactions at higher strain rates (𝒪⁢(1⁢0 2 −1⁢0 3 )⁢s −1 ) and larger overpressure magnitudes (𝒪⁡(1⁢0 2 )⁢kPa) within a biphasic soft porous material using Theory of Porous Media (TPM) at finite strain. Specifically, these constitutive modeling improvements are the following: (i) a more physically robust constitutive model for pore fluid seepage velocity via inclusion of pore fluid viscous stress, and (ii) a modified deformation-dependent-permeability model and updated hyperelastic constitutive model better suited for handling larger volumetric compressions and extensions. The novelty of the present work is mainly the contribution (i): inclusion of pore fluid viscous stress at higher strain-rate and large deformations, which requires 𝐶 1 continuity in the weak formulation, accomplished by employing Hermite cubic interpolation functions within a mixed nonlinear poromechanical finite element formulation. In (ii), the model is updated to weakly enforce solid phase incompressibility, such that this assumption is not violated numerically, which provides improved numerical stability for achieving larger overpressure magnitudes on 𝒪⁡(1⁢0 2 ) kPa, which were not achievable with the previous Kozeny–Carman model in Irwin et al. 2024. Also in (ii), the volumetric part of the solid skeleton free energy function is modified to ensure proper bounds on the solid skeleton Jacobian of deformation 𝐽 s related to incompressibility of the solid phase. Uniaxial strain, unidirectional flow examples at higher strain rates (𝒪⁢(1⁢0 2 −1⁢0 3 )⁢s −1 ) and larger deformations (up to 0.2 (or 20%) nominal axial strain) demonstrate the improved physical representation—and numerical stability—of these constitutive model improvements.

42 ENGINEERING↗
NASA NTRS1995

Aeroelasticity and structural optimization of composite helicopter rotor blades with swept tips

This report describes the development of an aeroelastic analysis capability for composite helicopter rotor blades with straight and swept tips, and its application to the simulation of helicopter vibration reduction through structural optimization. A new aeroelastic model is developed in this study which is suitable for composite rotor blades with swept tips in hover and in forward flight. The hingeless blade is modeled by beam type finite elements. A single finite element is used to model the swept tip. Arbitrary cross-sectional shape, generally anisotropic material behavior, transverse shears and out-of-plane warping are included in the blade model. The nonlinear equations of motion, derived using Hamilton's principle, are based on a moderate deflection theory. Composite blade cross-sectbnal properties are calculated by a separate linear, two-dimensional cross section analysis. The aerodynamic loads are obtained from quasi-steady, incompressible aerodynamics, based on an implicit formulation. The trim and steady state blade aeroelastic response are solved in a fully coupled manner. In forward flight, where the blade equations of motion are periodic, the coupled trim-aeroelastic response solution is obtained from the harmonic balance method. Subsequently, the periodic system is linearized about the steady state response, and its stability is determined from Floquet theory.

Yuan, K. A.↗
DOE OSTI2022

Droplet formation simulation using mixed finite elements

Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [“Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] using asymptotic analysis. This asymptotic approach to the Navier–Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference [Eggers and Dupont, “Drop formation in a one-dimensional approximation of the Navier–Stokes equation,” J. Fluid Mech. 262, 205–221 (1994)] and finite element method [Ambravaneswaran et al., “Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops,” Phys. Fluids 14, 2606–2621 (2002)]. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [“An experimental study of dynamics of drop formation,” Phys. Fluids 7, 1184–1203 (1995)]. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRSDate not supplied

FUN3D Mesh Adaptation Simulations of 4th AIAA High-Lift Prediction Workshop Case 1a

This paper describes in detail the FUN3D Mesh adaptation simulation results for Case 1a of the 4th AIAA High-Lift Prediction Workshop (HLPW-4). The Stabilized Finite Element library with FUN3D is used along with the Negative Spalart-Allmaras One-Equation Model to simulate fully turbulent free-air flow around the High Lift version of the Common Research Model at three flap settings. Meshes are adapted using multi-scale and goal-oriented metrics. The results are presented and the discussion focuses on the predicted differences between the three flap settings. Coefficients of lift, drag, and pitching moment are predicted within 7% of values measured in the QinetiQ 5m Wind Tunnel at each flap setting. Predicted surface pressures and skin friction distributions show inconsistent discrepancies with measurements across the flap settings. Consequently, flap increments for these figures of merit are not well predicted. These predictions are consistent with submissions to the HLPW-4 from other RANS-SA methods. Visualizations of Liutex vortex cores colored by angular velocity are presented as possible new quantitative and qualitative perspective for solution evaluation.

High-Lift Prediction Workshop↗
DOE OSTI2022

Between Harmonic Crystal and Glass: Solids with Dimpled Potential-Energy Surfaces Having Multiple Local Energy Minima

Solids with dimpled potential-energy surfaces are ubiquitous in nature and, typically, exhibit structural (elastic or phonon) instabilities. Dimpled potentials are not harmonic; thus, the conventional quasiharmonic approximation at finite temperatures fails to describe anharmonic vibrations in such solids. At sufficiently high temperatures, their crystal structure is stabilized by entropy; in this phase, a diffraction pattern of a periodic crystal is combined with vibrational properties of a phonon glass. As temperature is lowered, the solid undergoes a symmetry-breaking transition and transforms into a lower-symmetry phase with lower lattice entropy. Here, we identify specific features in the potential-energy surface that lead to such polymorphic behavior; we establish reliable estimates for the relative energies and temperatures associated with the anharmonic vibrations and the solid–solid symmetry-breaking phase transitions. We show that computational phonon methods can be applied to address anharmonic vibrations in a polymorphic solid at fixed temperature. To illustrate the ubiquity of this class of materials, we present a range of examples (elemental metals, a shape-memory alloy, and a layered charge-density-wave system); we show that our theoretical predictions compare well with known experimental data.

36 MATERIALS SCIENCE↗
NASA NTRS1986

Free vibration characteristics of multiple load path blades by the transfer matrix method

The determination of free vibrational characteristics is basic to any dynamic design, and these characteristics can form the basis for aeroelastic stability analyses. Conventional helicopter blades are typically idealized as single-load-path blades, and the transfer matrix method is well suited to analyze such blades. Several current helicopter dynamic programs employ transfer matrices to analyze the rotor blades. In this paper, however, the transfer matrix method is extended to treat multiple-load-path blades, without resorting to an equivalent single-load-path approximation. With such an extension, these current rotor dynamic programs which employ the transfer matrix method can be modified with relative ease to account for the multiple load paths. Unlike the conventional blades, the multiple-load-path blades require the introduction of the axial degree-of-freedom into the solution process to account for the differential axial displacements of the different load paths. The transfer matrix formulation is validated through comparison with the finite-element solutions.

Murthy, V. R.↗
NASA NTRS1991

Finite element computation of compressible flows with the SUPG formulation

Finite element computation of compressible Euler equations is presented in the context of the streamline-upwind/Petrov-Galerkin (SUPG) formulation. The SUPG formulation, which is based on adding stabilizing terms to the Galerkin formulation, is further supplemented with a shock capturing operator which addresses the difficulty in maintaining a satisfactory solution near discontinuities in the solution field. The shock capturing operator, which has been derived from work done in entropy variables for a similar operator, is shown to lead to an appropriate level of additional stabilization near shocks, without resulting in excessive numerical diffusion. An implicit treatment of the impermeable wall boundary condition is also presented. This treatment of the no-penetration condition offers increased stability for large Courant numbers, and accelerated convergence of the computations for both implicit and explicit applications. Several examples are presented to demonstrate the ability of this method to solve the equations governing compressible fluid flow.

Le Beau, G. J.↗
DOE OSTI2023

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. Here, the formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

42 ENGINEERING↗
NASA NTRS1998

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

Koppenhoefer, Kyle C.↗
DOE OSTI2023

On Practical Aspects of Variational Consistency in Contact Dynamics

Usage of contact mechanics methodologies is a pervasive modeling requirement in dynamic simulations. While for some trivial problems, solutions taken from analytical geometry are available, use of a finite element framework is common to achieve formulation generality. This work explores two dynamic contact formulations: one based on the traditional node-to-segment (NTS) approach, and a variationally consistent segment-to-segment (STS) mortar formulation. The NTS formulation employed here enforces the constraints kinematically (i.e., the interpenetration is enforced to the solver tolerance), whereas the mortar approach uses Lagrange multipliers to enforce the contact constraints. Both approaches are implemented in the open-source finite element framework Multiphysics Object-Oriented Simulation Environment (MOOSE). The results highlight two relevant contact-interface-related dynamic phenomena in finite element simulations. First, stabilization of contact constraints is discussed, taking into account the evolution of the total energy in a benchmark problem. Second, the influence of finite element discretization on both of the aforementioned contact formulations is analyzed by exercising a large-deformation example with continuous relative sliding. Variationally consistent contact approaches such as the mortar formulation lead to improved energy preservation and avoid spurious excitation of the system's frequencies. This is especially relevant in settings where inertia and vibrations are of importance.

42 ENGINEERING↗
NASA NTRSDate not supplied

Advanced Modeling of Control-Structure Interaction in Thrust Vector Control Systems

The Space Launch System (SLS) Core Stage (CS) Thrust Vector Control (TVC) system is comprised of 8 mechanical feedback Shuttle heritage Type III TVC actuators and four RS-25 engines, each attached to a Shuttle heritage gimbal block/bearing. Two actuators are used to move each engine in two planes perpendicular to one another (i.e., pitch and yaw). The TVC system design leverages hardware from the Space Shuttle program as well as new hardware designed specifically for the Core Stage. During the development of the SLS TVC system, a family of advanced dynamics models were developed to extend and compliment the simplified quasi-linear “simplex” model historically used for flight control design and stability analysis. The importance of these advanced models became increasingly evident after ambient and hot fire testing of the Core Stage, which revealed a number of findings associated with the dynamic response of the TVC integrated system. Test responses suggested that the TVC did not meet its performance specifications and its step and frequency responses exhibited unexpected departures from prior lab tests and modeled behavior. One driving factor for these results was a higher-than-expected degree of coupling between the TVC system, the engine dynamics, and the Core Stage structure. This paper is the third installment in a seven-paper series surveying the design, engineering, test validation, and flight performance of the Core Stage Thrust Vector Control system. In this paper, a new method of modeling rocket vehicle thrust vectoring servoelastic dynamics is presented. In this approach, the load dynamics are replaced by a detailed finite element model containing both the rigid body and elastic modes. A partitioning technique is used to compute the effective compliance from the modal data and obtain accurate simulation results using a reduced number of generalized coordinates. Coupled backup structure and nozzle attach compliance effects on multiple engines are captured in higher fidelity than with a spring approximation, eliciting novel effects due to the complex load paths involved in the Core Stage structure. Validation of the model is demonstrated using a variety of structural/modal, laboratory, and full-scale hot fire test data.

Launch Vehicles↗
NASA NTRSDate not supplied

Advanced Modeling of Control-Structure Interaction in Thrust Vector Control Systems

The Space Launch System (SLS) Core Stage (CS) Thrust Vector Control (TVC) system is comprised of 8 mechanical feedback Shuttle heritage Type III TVC actuators and four RS-25 engines, each attached to a Shuttle heritage gimbal block/bearing. Two actuators are used to move each engine in two planes perpendicular to one another (i.e., pitch and yaw). The TVC system design leverages hardware from the Space Shuttle program as well as new hardware designed specifically for the Core Stage. During the development of the SLS TVC system, a family of advanced dynamics models were developed to extend and compliment the simplified quasi-linear “simplex” model historically used for flight control design and stability analysis. The importance of these advanced models became increasingly evident after ambient and hot fire testing of the Core Stage, which revealed a number of findings associated with the dynamic response of the TVC integrated system. Test responses suggested that the TVC did not meet its performance specifications and its step and frequency responses exhibited unexpected departures from prior lab tests and modeled behavior. One driving factor for these results was a higher-than-expected degree of coupling between the TVC system, the engine dynamics, and the Core Stage structure. This paper is the third installment in a seven-paper series surveying the design, engineering, test validation, and flight performance of the Core Stage Thrust Vector Control system. In this paper, a new method of modeling rocket vehicle thrust vectoring servoelastic dynamics is presented. In this approach, the load dynamics are replaced by a detailed finite element model containing both the rigid body and elastic modes. A partitioning technique is used to compute the effective compliance from the modal data and obtain accurate simulation results using a reduced number of generalized coordinates. Coupled backup structure and nozzle attach compliance effects on multiple engines are captured in higher fidelity than with a spring approximation, eliciting novel effects due to the complex load paths involved in the Core Stage structure. Validation of the model is demonstrated using a variety of structural/modal, laboratory, and full-scale hot fire test data.

Launch Vehicles↗
NASA NTRS2000

Magnetic Field Suppression of Flow in Semiconductor Melt

One of the most promising approaches for the reduction of convection during the crystal growth of conductive melts (semiconductor crystals) is the application of magnetic fields. Current technology allows the experimentation with very intense static fields (up to 80 KGauss) for which nearly convection free results are expected from simple scaling analysis in stabilized systems (vertical Bridgman method with axial magnetic field). However, controversial experimental results were obtained. The computational methods are, therefore, a fundamental tool in the understanding of the phenomena accounting during the solidification of semiconductor materials. Moreover, effects like the bending of the isomagnetic lines, different aspect ratios and misalignments between the direction of the gravity and magnetic field vectors can not be analyzed with analytical methods. The earliest numerical results showed controversial conclusions and are not able to explain the experimental results. Although the generated flows are extremely low, the computational task is a complicated because of the thin boundary layers. That is one of the reasons for the discrepancy in the results that numerical studies reported. Modeling of these magnetically damped crystal growth experiments requires advanced numerical methods. We used, for comparison, three different approaches to obtain the solution of the problem of thermal convection flows: (1) Spectral method in spectral superelement implementation, (2) Finite element method with regularization for boundary layers, (3) Multiquadric method, a novel method with global radial basis functions, that is proven to have exponential convergence. The results obtained by these three methods are presented for a wide region of Rayleigh and Hartman numbers. Comparison and discussion of accuracy, efficiency, reliability and agreement with experimental results will be presented as well.

Fedoseyev, A. I.↗
NASA NTRS1990

General Rotorcraft Aeromechanical Stability Program (GRASP): Theory manual

The general rotorcraft aeromechanical stability program (GRASP) was developed to calculate aeroelastic stability for rotorcraft in hovering flight, vertical flight, and ground contact conditions. GRASP is described in terms of its capabilities and its philosophy of modeling. The equations of motion that govern the physical system are described, as well as the analytical approximations used to derive them. The equations include the kinematical equation, the element equations, and the constraint equations. In addition, the solution procedures used by GRASP are described. GRASP is capable of treating the nonlinear static and linearized dynamic behavior of structures represented by arbitrary collections of rigid-body and beam elements. These elements may be connected in an arbitrary fashion, and are permitted to have large relative motions. The main limitation of this analysis is that periodic coefficient effects are not treated, restricting rotorcraft flight conditions to hover, axial flight, and ground contact. Instead of following the methods employed in other rotorcraft programs. GRASP is designed to be a hybrid of the finite-element method and the multibody methods used in spacecraft analysis. GRASP differs from traditional finite-element programs by allowing multiple levels of substructure in which the substructures can move and/or rotate relative to others with no small-angle approximations. This capability facilitates the modeling of rotorcraft structures, including the rotating/nonrotating interface and the details of the blade/root kinematics for various types. GRASP differs from traditional multibody programs by considering aeroelastic effects, including inflow dynamics (simple unsteady aerodynamics) and nonlinear aerodynamic coefficients.

Hodges, Dewey H.↗
NASA NTRS1992

Time simulation of flutter with large stiffness changes

Time simulation of flutter, involving large local structural changes, is formulated with a state-space model that is based on a relatively small number of generalized coordinates. Free-free vibration modes are first calculated for a nominal finite-element model with relatively large fictitious masses located at the area of structural changes. A low-frequency subset of these modes is then transformed into a set of structural modal coordinates with which the entire simulation is performed. These generalized coordinates and the associated oscillatory aerodynamic force coefficient matrices are used to construct an efficient time-domain, state-space model for a basic aeroelastic case. The time simulation can then be performed by simply changing the mass, stiffness, and damping coupling terms when structural changes occur. It is shown that the size of the aeroelastic model required for time simulation with large structural changes at a few apriori known locations is similar to that required for direct analysis of a single structural case. The method is applied to the simulation of an aeroelastic wind-tunnel model. The diverging oscillations are followed by the activation of a tip-ballast decoupling mechanism that stabilizes the system but may cause significant transient overshoots.

Karpel, Mordechay↗
NASA NTRS1992

Time simulation of flutter with large stiffness changes

Time simulation of flutter, involving large local structural changes, is formulated with a state-space model that is based on a relatively small number of generalized coordinates. Free-free vibration modes are first calculated for a nominal finite-element model with relatively large fictitious masses located at the area of structural changes. A low-frequency subset of these modes is then transformed into a set of structural modal coordinates with which the entire simulation is performed. These generalized coordinates and the associated oscillatory aerodynamic force coefficient matrices are used to construct an efficient time-domain, state-space model for basic aeroelastic case. The time simulation can then be performed by simply changing the mass, stiffness and damping coupling terms when structural changes occur. It is shown that the size of the aeroelastic model required for time simulation with large structural changes at a few a priori known locations is similar to that required for direct analysis of a single structural case. The method is applied to the simulation of an aeroelastic wind-tunnel model. The diverging oscillations are followed by the activation of a tip-ballast decoupling mechanism that stabilizes the system but may cause significant transient overshoots.

Karpel, M.↗
NASA NTRSDate not supplied

Dynamic Aeroelastic Flight Dynamic Modeling of Mach 0.745 Transonic Truss-Braced Wing

This paper presents a modeling approach for dynamic aeroelastic flight dynamic analysis of the Mach0.745 Transonic Truss-Braced Wing. The modeling approach is based on a transonic correction method to correct the Theodorsen’s theory for transonic flow. CFD unsteady Reynolds-averaged Navier-Stokes equations (RANS) simulations are conducted using FUN3D solver for a series of wing sections from the Mach 0.745Transonic Truss-Braced Wing in pitch and plunge oscillations. Unsteady lift and pitching moment coefficients are obtained and used to develop the correction terms in the Theodorsen’s theory to account for transonic aerodynamics. The unsteady lift and pitching moment derivatives with respect to the unsteady angle of attack are obtained as functions of the reduced frequency. These derivatives are used to compute the unsteady lift and pitching moment contributions by the angle of attack, pitch rate, and roll rate. They are then approximated using a frequency-domain regression to obtain the dynamic stability derivatives for the Mach 0.745 Transonic Truss-Braced Wing. The structural dynamic mode shapes of the Mach 0.745 TTBW are extracted from a NAS-TRAN finite-element model. These mode shapes are used to compute the generalized unsteady aerodynamic forces. The aerodynamic mass, damping, and stiffness and the aerodynamic lag states are constructed to couple the dynamic aeroelastic contribution to the flight dynamic model of the Mach 0.745 Transonic Truss-Braced Wing. The coupled dynamic aeroelastic flight dynamic equations of motion are formulated. The eigenvalues of the coupled system are computed. All the flight dynamic modes and structural dynamic modes are stable at Match 0.745. The effect of transonic aerodynamics generally causes all the dynamic modes to have lower damping values

Flight Dynamics↗
NASA NTRS2014

Mean Flow Augmented Acoustics in Rocket Systems

Oscillatory motion in solid rocket motors and liquid engines has long been a subject of concern. Many rockets display violent fluctuations in pressure, velocity, and temperature originating from the complex interactions between the combustion process and gas dynamics. The customary approach to modeling acoustic waves inside a rocket chamber is to apply the classical inhomogeneous wave equation to the combustion gas. The assumption of a linear, non-dissipative wave in a quiescent fluid remains valid while the acoustic amplitudes are small and local gas velocities stay below Mach 0.2. The converging section of a rocket nozzle, where gradients in pressure, density, and velocity become large, is a notable region where this approach is not applicable. The expulsion of unsteady energy through the nozzle of a rocket is identified as the predominate source of acoustic damping for most rocket systems. An accurate model of the acoustic behavior within this region where acoustic modes are influenced by the presence of a steady mean flow is required for reliable stability predictions. Recently, an approach to address nozzle damping with mean flow effects was implemented by French [1]. This new approach extends the work originated by Sigman and Zinn [2] by solving the acoustic velocity potential equation (AVPE) formulated by perturbing the Euler equations [3]. The acoustic velocity potential (psi) describing the acoustic wave motion in the presence of an inhomogeneous steady high-speed flow is defined by, (del squared)(psi) − (lambda/c)(exp 2)(psi) − M(dot)[M(dot)(del)(del(psi))] − 2(lambda(M/c) + (M(dot)del(M))(dot)del(psi)−2(lambda)(psi)[M(dot)del(1/c)]=0 (1) with M as the Mach vector, c as the speed of sound, and lambda as the complex eigenvalue. French apply the finite volume method to solve the steady flow field within the combustion chamber and nozzle with inviscid walls. The complex eigenvalues and eigenvector are determined with the use of the ARPACK eigensolver. The present study employs the COMSOL Multphysics framework to solve the coupled eigenvalue problem using the finite element approach. The study requires one way coupling of the CFD High Mach Number Flow (HMNF) and mathematics module. The HMNF module evaluated the gas flow inside of a solid rocket motor using St. Robert's law modeling solid propellant burn rate, slip boundary conditions, and the supersonic outflow condition. Results from the HMNF model are used by the coefficient form of the mathematics module to determine the eigenvalues of the AVPE. The mathematics model is truncated at the nozzle sonic line, where a zero flux boundary condition is self-satisfying. The remaining boundaries are modeled with a zero flux boundary condition, assuming zero acoustic absorption on all surfaces. Pertinent results from these analyses are the complex valued eigenvalue and eigenvectors. Comparisons are made to the French results to evaluate the modeling approach. A comparison of the French results with that of the present analysis is displayed in figures 1 and 2, respectively. The graphic shows the first tangential eigenvector's real (a) and imaginary (b) values.

Fischbach, Sean R.↗