Results for “finite volume discretization” — Page 12

NASA NTRS1991

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems. Task 2: Unsteady ducted propfan analysis computer program users manual

The primary objective of this study was the development of a time-dependent three-dimensional Euler/Navier-Stokes aerodynamic analysis to predict unsteady compressible transonic flows about ducted and unducted propfan propulsion systems at angle of attack. The computer codes resulting from this study are referred to as Advanced Ducted Propfan Analysis Codes (ADPAC). This report is intended to serve as a computer program user's manual for the ADPAC developed under Task 2 of NASA Contract NAS3-25270, Unsteady Ducted Propfan Analysis. Aerodynamic calculations were based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. A time-accurate implicit residual smoothing operator was utilized for unsteady flow predictions. For unducted propfans, a single H-type grid was used to discretize each blade passage of the complete propeller. For ducted propfans, a coupled system of five grid blocks utilizing an embedded C-grid about the cowl leading edge was used to discretize each blade passage. Grid systems were generated by a combined algebraic/elliptic algorithm developed specifically for ducted propfans. Numerical calculations were compared with experimental data for both ducted and unducted propfan flows. The solution scheme demonstrated efficiency and accuracy comparable with other schemes of this class.

Hall, Edward J.↗

NASA NTRS1992

A finite element boundary integral formulation for radiation and scattering by cavity antennas using tetrahedral elements

A hybrid finite element boundary integral formulation is developed using tetrahedral and/or triangular elements for discretizing the cavity and/or aperture of microstrip antenna arrays. The tetrahedral elements with edge based linear expansion functions are chosen for modeling the volume region and triangular elements are used for discretizing the aperture. The edge based expansion functions are divergenceless thus removing the requirement to introduce a penalty term and the tetrahedral elements permit greater geometrical adaptability than the rectangular bricks. The underlying theory and resulting expressions are discussed in detail together with some numerical scattering examples for comparison and demonstration.

Gong, J.↗

NASA NTRS1992

The effects of uneven fiber spacing on thermal residual stresses in a unidirectional SCS-6Ti-15-3 laminate

High residual stresses develop in SCS-6/Ti-15-3 composites during cooldown from the fabrication temperature; these residual stresses can effect the mechanical and physical properties of the composite. Discrete fiber-matrix finite element models were used to study the residual stresses due to the temperature change during the fabrication process, including the effects of uneven fiber spacing, the free surface, and increased fiber volume fractions. To accurately model the effects of the free surface, it is only necessary to model one fiber through the thickness. Below the first ply, the analysis predicts stress distributions that are identical to the infinite array predictions. For uneven fiber space less then 0.042 mm in an interior ply, the maximum hoop stress was predicted to occur between fibers within a ply and to increase as the fiber spacing decreased. The maximum hoop stress correlated well with the observed radial cracking between fibers. For the case of touching fibers, the analysis predicted tensile radial stresses at the fiber-matrix debonding during the fabrication cooldown. Identical trends were predicted for uneven fiber spacing in surface plies with slightly greater values of maximum stresses. The analysis predicted matrix yielding to occur upon cooldown when the edge-to-edge fiber spacing was less than or equal to 0.022 mm. The stress distributions predicted for increasing fiber volume fractions were similar to those predicted for decreasing the fiber spacing for two adjacent fibers within a ply.

Bigelow, C. A.↗

NASA NTRS1992

The effects of uneven fiber spacing on thermal residual stresses in a unidirectional SCS-6/Ti-15-3 laminate

High residual stresses develop in SCS-6/Ti-15-3 composites during cooldown from the fabrication temperature; these residual stresses can effect the mechanical and physical properties of the composite. Discrete fiber-matrix finite element models were used to study the residual stresses due to the temperature change during the fabrication process, including the effects of uneven fiber spacing, the free surface, and increased fiber volume fractions. To accurately model the effects of the free surface, it is only necessary to model one fiber through the thickness. Below the first ply, the analysis predicts stress distributions that are identical to the infinite array predictions. For uneven fiber space less than 0.042 mm in an interior ply, the maximum hoop stress was predicted to occur between fibers within a ply and to increase as the fiber spacing decreased. The maximum hoop stress correlated well with the observed radial cracking between fibers. For the case of touching fibers, the analysis predicted tensile radial stresses at the fiber-matrix debonding during the fabrication cooldown. Identical trends were predicted for uneven fiber spacing in surface plies with slightly greater values of maximum stresses. The analysis predicted matrix yielding to occur upon cooldown when the edge-to-edge fiber spacing was less than or equal to 0.022 mm. The stress distributions preidcted for increasing fiber volume fractions were similar to those predicted for decreasing the fiber spacing for two adjacent fibers within a ply.

Bigelow, Catherine A.↗

NASA NTRS2017

Stabilized Finite Elements in FUN3D

A Streamlined Upwind Petrov-Galerkin (SUPG) stabilized finite-element discretization has been implemented as a library into the FUN3D unstructured-grid flow solver. Motivation for the selection of this methodology is given, details of the implementation are provided, and the discretization for the interior scheme is verified for linear and quadratic elements by using the method of manufactured solutions. A methodology is also described for capturing shocks, and simulation results are compared to the finite-volume formulation that is currently the primary method employed for routine engineering applications. The finite-element methodology is demonstrated to be more accurate than the finite-volume technology, particularly on tetrahedral meshes where the solutions obtained using the finite-volume scheme can suffer from adverse effects caused by bias in the grid. Although no effort has been made to date to optimize computational efficiency, the finite-element scheme is competitive with the finite-volume scheme in terms of computer time to reach convergence.

Anderson, W. Kyle↗

NASA NTRSDate not supplied

Two-Dimensional Network Flow Modeling of No-Vent Tank Filling of A Cryogenic Tank With Thermo-Dynamic Vent System Assisted Injector

A finite volume-based network flow modeling tool, Generalized Fluid System Simulation Program (GFSSP), was used to simulate the no-vent filling of a cryogenic tank with a thermodynamic vent system assisted injector. The tank was discretized into nodes in the radial and axial directions to calculate two-dimensional axisymmetric flow with heat transfer between the solid wall and fluid. The nodes are connected by branches where momentum equations are solved to calculate the flowrates. The mass and energy conservation equations and the equation of state are solved to calculate pressure, temperature, and resident mass at the nodes. The energy conservation equations at the solid nodes were solved to calculate the temperature of the solid wall. The system of equations was solved by a combination of successive substitution and simultaneous Newton-Raphson method. The numerical model accounts for a) different regimes of pool boiling heat transfer, b) condensation of vapor around the spray droplets, c) condensation of vapor at the interface of the cooled injector and ullage, and d) condensation at the liquid-vapor interphase. The numerical predictions of tank pressure, filling rate, and wall temperatures were compared with the test data. The paper also presents the effect of nodal discretization by comparing the results of the 1D and 2D models and examines the effect of the droplet diameter on tank filling.

Nodal Model↗

NASA NTRS2023

Two-Dimensional Network Flow Modeling of No-Vent Tank Filling of A Cryogenic Tank With Thermo-Dynamic Vent System Assisted Injector

A finite volume-based network flow modeling tool, Generalized Fluid System Simulation Program (GFSSP), was used to simulate the no-vent filling of a cryogenic tank with a thermodynamic vent system assisted injector. The tank was discretized into nodes in the radial and axial directions to calculate two-dimensional axisymmetric flow with heat transfer between the solid wall and fluid. The nodes are connected by branches where momentum equations are solved to calculate the flowrates. The mass and energy conservation equations and the equation of state are solved to calculate pressure, temperature, and resident mass at the nodes. The energy conservation equations at the solid nodes were solved to calculate the temperature of the solid wall. The system of equations was solved by a combination of successive substitution and simultaneous Newton-Raphson method. The numerical model accounts for a) different regimes of pool boiling heat transfer, b) condensation of vapor around the spray droplets, c) condensation of vapor at the interface of the cooled injector and ullage, and d) condensation at the liquid-vapor interphase. The numerical predictions of tank pressure, filling rate, and wall temperatures were compared with the test data. The paper also presents the effect of nodal discretization by comparing the results of the 1D and 2D models and examines the effect of the droplet diameter on tank filling.

Nodal Model↗

NASA NTRS2007

Simulation of a Periodic Jet in a Crossflow with a RANS Solver Using an Unstructured Grid

A second-order unstructured-grid code, developed and used primarily for steady aerodynamic simulations, is applied to the synthetic jet in a cross flow. The code, FUN3D, is a vertex-centered finite-volume method originally developed by Anderson[1, 2], and is currently supported by members of the Fast Adaptive Aerospace Tools team at NASA Langley. Used primarily for design[3] and analysis[4] of steady aerodynamic configurations, FUN3D incorporates a discrete adjoint capability, and supports parallel computations using MPI. A detailed description of the FUN3D code can be found in the references given above. The code is under continuous development and contains a variety of flux splitting algorithms for the inviscid terms, two methods for computing gradients, several turbulence models, and several solution methodologies; all in varying states of development. Only the most robust and reliable components, based on experiences with steady aerodynamic simulations, were employed in this work. As applied in this work, FUN3D solves the Reynolds averaged Navier-Stokes equations using the one equation turbulence model of Spalart and Allmaras[5]. The spatial discretization is formed on unstructured meshes using a vertex-centered approach. The inviscid terms are evaluated by a flux-difference splitting formulation using least-squares reconstruction and Roe-type approximate Riemann fluxes. Green-Gauss gradient evaluations are used for viscous and turbulence modeling terms. The discrete spatial operator is combined with a backward time operator which is then solved iteratively using point or line Gauss-Seidel and local time stepping in a pseudo time. For steady flows, the physical time step is set to infinity and the pseudo time step is ramped up with the iteration count. A second-order backward in time operator is used for time accurate flows with 20 to 50 steps in the pseudo time applied at each physical time step. For this effort, FUN3D was modified to support spatially varying boundary and initial conditions, and unsteady boundary conditions. Also, a specialized in/out flow boundary condition was implemented to model the action of the diaphragm. This boundary condition is described below in more detail. The grids were generated using the internally developed codes GridEX[6] for meshing the surfaces and inviscid regions of the domain, and for CAD access; and MesherX[7] for meshing the viscous regions. Grid spacing in on the surfaces and in the inviscid regions are indirectly controlled by specifying sources. The viscous layers are generated using an advancing layer technique. MeshersX allows the user to control the spatial variation of the first step off the surface, growth rates, and the termination criterion by providing small problem dependent subroutines.

Atkins, H. L.↗

NASA NTRS2018

High-Order Stabilized Finite Elements on Dynamic Meshes

The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered.

Anderson, W. Kyle↗

NASA NTRS1990

An upwind-biased, point-implicit relaxation algorithm for viscous, compressible perfect-gas flows

An upwind-biased, point-implicit relaxation algorithm for obtaining the numerical solution to the governing equations for three-dimensional, viscous, compressible, perfect-gas flows is described. The algorithm is derived using a finite-volume formulation in which the inviscid components of flux across cell walls are described with Roe's averaging and Harten's entropy fix with second-order corrections based on Yee's Symmetric Total Variation Diminishing scheme. Viscous terms are discretized using central differences. The relaxation strategy is well suited for computers employing either vector or parallel architectures. It is also well suited to the numerical solution of the governing equations on unstructured grids. Because of the point-implicit relaxation strategy, the algorithm remains stable at large Courant numbers without the necessity of solving large, block tri-diagonal systems. Convergence rates and grid refinement studies are conducted for Mach 5 flow through an inlet with a 10 deg compression ramp and Mach 14 flow over a 15 deg ramp. Predictions for pressure distributions, surface heating, and aerodynamics coefficients compare well with experiment data for Mach 10 flow over a blunt body.

Gnoffo, Peter A.↗

NASA NTRS1980

Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics, production version (SOUSSA-P 1.1). Volume 1: Theoretical manual

Recent developments of the Green's function method and the computer program SOUSSA (Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed and summarized. Applying the Green's function method to the fully unsteady (transient) potential equation yields an integro-differential-delay equation. With spatial discretization by the finite-element method, this equation is approximated by a set of differential-delay equations in time. Time solution by Laplace transform yields a matrix relating the velocity potential to the normal wash. Premultiplying and postmultiplying by the matrices relating generalized forces to the potential and the normal wash to the generalized coordinates one obtains the matrix of the generalized aerodynamic forces. The frequency and mode-shape dependence of this matrix makes the program SOUSSA useful for multiple frequency and repeated mode-shape evaluations.

Morino, L.↗

NASA NTRS2008

A New Concurrent Multiscale Methodology for Coupling Molecular Dynamics and Finite Element Analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.

Yamakov, Vesselin↗

NASA NTRS2008

An Embedded Statistical Method for Coupling Molecular Dynamics and Finite Element Analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.

Saether, E.↗

NASA NTRS1987

Three-dimensional unsteady Euler solutions for propfans and counter-rotating propfans in transonic flow

An Euler code designed for computing the unsteady, three-dimensional, transonic flow about single-rotating and counter-rotating propfans using dynamic blocked-grids is presented. The algorithm is a finite volume, flux-split, upwind, implicit scheme and solves the equations which have been written in a time-dependent curvilinear coordinate system. Relative motion of the blades for counter-rotating configurations is handled by requiring that grid lines be aligned after each discrete rotation of fore and aft rotor grid blocks. The method by which information is passed across block interfaces, as well as how downstream characteristic outflow boundary conditions which enforce simple radial equilibrium are implemented, is discussed. Comparisons of computed flow-field parameters and propfan performance with experimental data indicate good overall agreement between predictions and measurements.

Whitfield, D. L.↗

NASA NTRS1993

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems. Task 5: Unsteady counterrotation ducted propfan analysis. Computer program user's manual

The primary objective of this study was the development of a time-marching three-dimensional Euler/Navier-Stokes aerodynamic analysis to predict steady and unsteady compressible transonic flows about ducted and unducted propfan propulsion systems employing multiple blade rows. The computer codes resulting from this study are referred to as ADPAC-AOACR (Advanced Ducted Propfan Analysis Codes-Angle of Attack Coupled Row). This report is intended to serve as a computer program user's manual for the ADPAC-AOACR codes developed under Task 5 of NASA Contract NAS3-25270, Unsteady Counterrotating Ducted Propfan Analysis. The ADPAC-AOACR program is based on a flexible multiple blocked grid discretization scheme permitting coupled 2-D/3-D mesh block solutions with application to a wide variety of geometries. For convenience, several standard mesh block structures are described for turbomachinery applications. Aerodynamic calculations are based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. Steady flow predictions are accelerated by a multigrid procedure. Numerical calculations are compared with experimental data for several test cases to demonstrate the utility of this approach for predicting the aerodynamics of modern turbomachinery configurations employing multiple blade rows.

Hall, Edward J.↗

NASA NTRS1993

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems. Task 5: Unsteady counterrotation ducted propfan analysis

The primary objective of this study was the development of a time-marching three-dimensional Euler/Navier-Stokes aerodynamic analysis to predict steady and unsteady compressible transonic flows about ducted and unducted propfan propulsion systems employing multiple blade rows. The computer codes resulting from this study are referred to as ADPAC-AOAR\CR (Advanced Ducted Propfan Analysis Codes-Angle of Attack Coupled Row). This document is the final report describing the theoretical basis and analytical results from the ADPAC-AOACR codes developed under task 5 of NASA Contract NAS3-25270, Unsteady Counterrotating Ducted Propfan Analysis. The ADPAC-AOACR Program is based on a flexible multiple blocked grid discretization scheme permitting coupled 2-D/3-D mesh block solutions with application to a wide variety of geometries. For convenience, several standard mesh block structures are described for turbomachinery applications. Aerodynamic calculations are based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. Steady flow predictions are accelerated by a multigrid procedure. Numerical calculations are compared with experimental data for several test cases to demonstrate the utility of this approach for predicting the aerodynamics of modern turbomachinery configurations employing multiple blade rows.

Hall, Edward J.↗

NASA NTRS1996

Numerical Modeling of Spray Combustion with an Unstructured-Grid Method

The present unstructured-grid method follows strictly the basic finite volume forms of the conservation laws of the governing equations for the entire flow domain. High-order spatially accurate formulation has been employed for the numerical solutions of the Navier-Stokes equations. A two-equation k-epsilon turbulence model is also incorporated in the unstructured-grid solver. The convergence of the resulted linear algebraic equation is accelerated with preconditioned Conjugate Gradient method. A statistical spray combustion model has been incorporated into the present unstructured-grid solver. In this model, spray is represented by discrete particles, rather than by continuous distributions. A finite number of computational particles are used to predict a sample of total population of particles. Particle trajectories are integrated using their momentum and motion equations and particles exchange mass, momentum and energy with the gas within the computational cell in which they are located. The interaction calculations are performed simultaneously and eliminate global iteration for the two-phase momentum exchange. A transient spray flame in a high pressure combustion chamber is predicted and then the solution of liquid-fuel combusting flow with a rotating cup atomizer is presented and compared with the experimental data. The major conclusion of this investigation is that the unstructured-grid method can be employed to study very complicated flow fields of turbulent spray combustion. Grid adaptation can be easily achieved in any flow domain such as droplet evaporation and combustion zone. Future applications of the present model can be found in the full three-dimensional study of flow fields of gas turbine and liquid propulsion engine combustion chambers with multi-injectors.

Shang, H. M.↗

NASA NTRS2013

Toward Verification of USM3D Extensions for Mixed Element Grids

The unstructured tetrahedral grid cell-centered finite volume flow solver USM3D has been recently extended to handle mixed element grids composed of hexahedral, prismatic, pyramidal, and tetrahedral cells. Presently, two turbulence models, namely, baseline Spalart-Allmaras (SA) and Menter Shear Stress Transport (SST), support mixed element grids. This paper provides an overview of the various numerical discretization options available in the newly enhanced USM3D. Using the SA model, the flow solver extensions are verified on three two-dimensional test cases available on the Turbulence Modeling Resource website at the NASA Langley Research Center. The test cases are zero pressure gradient flat plate, planar shear, and bump-inchannel. The effect of cell topologies on the flow solution is also investigated using the planar shear case. Finally, the assessment of various cell and face gradient options is performed on the zero pressure gradient flat plate case.

Pandya, Mohagna J.↗