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NASA NTRS2026

A Modular Conjugate Heat Transfer Optimization Framework for Thermal Management of Electric Aircraft

Conjugate heat transfer (CHT) analysis and optimization is a powerful method for improving thermal management, as it simultaneously resolves the temperature distribution in both fluid and solid domains. This paper presents a modular, discrete adjoint-based CHT optimization capability integrated within the OpenMDAO/MPhys framework. A unique feature of the proposed framework is its flexibility to extend to multidisciplinary optimization, including aero-structural-thermal applications. The fluid domain is modeled using a finite-volume Computational Fluid Dynamics (CFD) solver, and the solid domain with a conduction heat transfer solver. A mixed Neumann-Dirichlet boundary condition is developed to enable full submersion of the solid geometry within the fluid domain, while ensuring consistent temperature and heat flux coupling at the CHT interface. Gradient-based optimization is performed; the gradients are efficiently computed using the discrete adjoint solvers implemented in DAFoam. To demonstrate the method, this paper considers two cases related to electric aircraft thermal management: a U-bend heat exchanger and an actively cooled battery pack. The U-bend case aims to minimize pressure loss while maximizing heat flux by changing the pipe geometry. The optimized design reduces pressure loss by 52.7% and increases total heat flux by 2.3%. In the battery pack case, a 3-by-3 cell configuration is cooled by ambient airflow, with constant heat generation prescribed in the cells. The battery casing shape serves as the design variable, and the objective function is a weighted sum of pressure loss and pack weight, subject to a maximum temperature constraint. The optimized design achieves a 44.6% reduction in pressure loss and a 1.5% reduction in weight, while satisfying the thermal constraint. To ensure the reliability of the optimized designs, this study validates coarse-mesh, steady-state predictions against fine-mesh unsteady simulations, demonstrating consistency within acceptable errors. This work demonstrates the potential of the developed framework to enable rapid, high-fidelity design of thermal management systems for electric aircraft.

heat transfer↗

NASA NTRS1985

Accuracy of schemes with nonuniform meshes for compressible fluid flows

The accuracy of the space discretization for time-dependent problems when a nonuniform mesh is used is considered. Many schemes reduce to first-order accuracy while a popular finite volume scheme is even inconsistent for general grids. This accuracy is based on physical variables. However, when accuracy is measured in computational variables then second-order accuracy can be obtained. This is meaningful only if the mesh accurately reflects the properties of the solution. In addition, the stability properties of some improved accurate schemes are analyzed and it can be shown that they also allow for larger time steps when Runge-Kutta type methods are used to advance in time.

Turkel, E.↗

NASA NTRS1986

Accuracy of schemes with nonuniform meshes for compressible fluid flows

Turkel, Eli↗

NASA NTRSDate not supplied

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

This paper, contains discussion on the implementation of new diffusion schemes for Svard’s Eulerian flow (EF) governing equations. It contains an alpha-damping type scheme and a new hyperbolic Eulerian flow (HEF) discretization paralleling previous work on hyperbolic Navier-Stokes (HNS). It shows these equations have simplified hyperbolic discretizations due to the simplicity of the new stress tensor and compares these schemes to established Navier-Stokes (NS) discretizations to verify the predictive utilities of such diffusion schemes.

Computational Fluid Dynamics↗

NASA NTRS2024

Finite-Volume Diffusion Schemes for Svard's Eulerian Governing Equations

Computational Fluid Dynamics↗

NASA NTRS2002

Multi-dimensional Upwind Fluctuation Splitting Scheme with Mesh Adaption for Hypersonic Viscous Flow

A multi-dimensional upwind fluctuation splitting scheme is developed and implemented for two-dimensional and axisymmetric formulations of the Navier-Stokes equations on unstructured meshes. Key features of the scheme are the compact stencil, full upwinding, and non-linear discretization which allow for second-order accuracy with enforced positivity. Throughout, the fluctuation splitting scheme is compared to a current state-of-the-art finite volume approach, a second-order, dual mesh upwind flux difference splitting scheme (DMFDSFV), and is shown to produce more accurate results using fewer computer resources for a wide range of test cases. A Blasius flat plate viscous validation case reveals a more accurate upsilon-velocity profile for fluctuation splitting, and the reduced artificial dissipation production is shown relative to DMFDSFV. Remarkably, the fluctuation splitting scheme shows grid converged skin friction coefficients with only five points in the boundary layer for this case. The second half of the report develops a local, compact, anisotropic unstructured mesh adaptation scheme in conjunction with the multi-dimensional upwind solver, exhibiting a characteristic alignment behavior for scalar problems. The adaptation strategy is extended to the two-dimensional and axisymmetric Navier-Stokes equations of motion through the concept of fluctuation minimization.

Wood, William A., III↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

The paper describes an approach to treating initial-boundary-value problems by finite volume methods in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second-order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and ADI methods.

Rose, Milton E.↗

NASA NTRS1994

New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems

A new numerical discretization method for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is motivated by several important physical/numerical considerations and designed to avoid several key limitations of the above traditional methods. As a result of the above considerations, a set of key principles for the design of numerical schemes was put forth in a previous report. These principles were used to construct several numerical schemes that model a 1-D time-dependent convection-diffusion equation. These schemes were then extended to solve the time-dependent Euler and Navier-Stokes equations of a perfect gas. It was shown that the above schemes compared favorably with the traditional schemes in simplicity, generality, and accuracy. In this report, the 2-D versions of the above schemes, except the Navier-Stokes solver, are constructed using the same set of design principles. Their constructions are simplified greatly by the use of a nontraditional space-time mesh. Its use results in the simplest stencil possible, i.e., a tetrahedron in a 3-D space-time with a vertex at the upper time level and other three at the lower time level. Because of the similarity in their design, each of the present 2-D solvers virtually shares with its 1-D counterpart the same fundamental characteristics. Moreover, it is shown that the present Euler solver is capable of generating highly accurate solutions for a famous 2-D shock reflection problem. Specifically, both the incident and the reflected shocks can be resolved by a single data point without the presence of numerical oscillations near the discontinuity.

Chang, Sin-Chung↗

NASA NTRS2011

Numerically Exact Computer Simulations of Light Scattering by Densely Packed, Random Particulate Media

Direct computer simulations of electromagnetic scattering by discrete random media have become an active area of research. In this progress review, we summarize and analyze our main results obtained by means of numerically exact computer solutions of the macroscopic Maxwell equations. We consider finite scattering volumes with size parameters in the range, composed of varying numbers of randomly distributed particles with different refractive indices. The main objective of our analysis is to examine whether all backscattering effects predicted by the low-density theory of coherent backscattering (CB) also take place in the case of densely packed media. Based on our extensive numerical data we arrive at the following conclusions: (i) all backscattering effects predicted by the asymptotic theory of CB can also take place in the case of densely packed media; (ii) in the case of very large particle packing density, scattering characteristics of discrete random media can exhibit behavior not predicted by the low-density theories of CB and radiative transfer; (iii) increasing the absorptivity of the constituent particles can either enhance or suppress typical manifestations of CB depending on the particle packing density and the real part of the refractive index. Our numerical data strongly suggest that spectacular backscattering effects identified in laboratory experiments and observed for a class of high-albedo Solar System objects are caused by CB.

Dlugach, Janna M.↗

NASA NTRS1989

Compact finite volume methods for the diffusion equation

An approach to treating initial-boundary value problems by finite volume methods is described, in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, it is possible to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl = 0 and curl grad = 0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and alternating direction implicit (ADI) methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.

Rose, Milton E.↗

NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗

NASA NTRS1978

A finite volume method for calculating transonic potential flow around wings from the pressure minimum integral

Analysis of the pressure minimum integral in the calculation of three-dimensional potential flow around wings makes it possible to use non-rectangular mesh networks for distributing the three-dimensional potential into discrete points. The method is comparatively easily expanded to the treatment of realistic airplane configurations. Shock-pressure affected pressure distributions on any wings are determined with accuracy using this method.

Eberle, A.↗

NASA NTRS1989

Computational simulation of damping in composite structures

A computational methodology is developed for the prediction of passive damping in composite structures. The method involves multiple levels of damping modeling by integrating micromechanics, laminate, and structural damping theories. The effects of temperature and moisture on structural damping are included. The simulation of damping in the structural level is accomplished with finite-element discretization. Applications are performed on graphite/epoxy composite beams, plates, and shells to illustrate the methodology. Additional parametric studies demonstrate the variation of structural modal damping with respect to ply angles, fiber volume ratio, and temperature.

Saravanos, D. A.↗

NASA NTRS1991

Investigation of Advanced Counterrotation Blade Configuration Concepts for High Speed Turboprop Systems. Task 2: Unsteady Ducted Propfan Analysis

The primary objective was the development of a time dependent 3-D Euler/Navier-Stokes aerodynamic analysis to predict unsteady compressible transonic flows about ducted and unducted propfan propulsion systems at angle of attack. The resulting computer codes are referred to as Advanced Ducted Propfan Analysis Codes (ADPAC). A computer program user's manual is presented for the ADPAC. 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 used 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 flows.

Hall, Edward J.↗

NASA NTRS1991

Computational simulation of damping in composite structures

Saravanos, D. A.↗

NASA NTRS1996

Task 7: ADPAC User's Manual

The overall objective of this study was to develop a 3-D numerical analysis for compressor casing treatment flowfields. The current version of the computer code resulting from this study is referred to as ADPAC (Advanced Ducted Propfan Analysis Codes-Version 7). This report is intended to serve as a computer program user's manual for the ADPAC code developed under Tasks 6 and 7 of the NASA Contract. The ADPAC program is based on a flexible multiple- block grid discretization scheme permitting coupled 2-D/3-D mesh block solutions with application to a wide variety of geometries. 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. An iterative implicit algorithm is available for rapid time-dependent flow calculations, and an advanced two equation turbulence model is incorporated to predict complex turbulent flows. The consolidated code generated during this study is capable of executing in either a serial or parallel computing mode from a single source code. Numerous examples are given in the form of test cases to demonstrate the utility of this approach for predicting the aerodynamics of modem turbomachinery configurations.

Hall, E. J.↗

NASA NTRS1999

ADPAC v1.0: User's Manual

The overall objective of this study was to evaluate the effects of turbulence models in a 3-D numerical analysis on the wake prediction capability. The current version of the computer code resulting from this study is referred to as ADPAC v7 (Advanced Ducted Propfan Analysis Codes -Version 7). This report is intended to serve as a computer program user's manual for the ADPAC code used and modified under Task 15 of NASA Contract NAS3-27394. The ADPAC program is based on a flexible multiple-block and discretization scheme permitting coupled 2-D/3-D mesh block solutions with application to a wide variety of geometries. 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. Turbulence models now available in the ADPAC code are: a simple mixing-length model, the algebraic Baldwin-Lomax model with user defined coefficients, the one-equation Spalart-Allmaras model, and a two-equation k-R model. The consolidated ADPAC code is capable of executing in either a serial or parallel computing mode from a single source code.

Hall, Edward J.↗

NASA NTRSDate not supplied

Extending an Economical Third-Order Inviscid Nodal-Gradient Cell-Centered Finite-Volume Method to Mixed-Element Grids

In this paper, we extend an economical third-order nodal-gradient cell-centered finite-volume method, originally developed for tetrahedral grids, to mixed-element grids. It is shown that the efficient quadratic interpolation formula essential to eliminating second derivatives from a third-order accurate discretization can be easily extended to an arbitrary cell type. For the surface flux integration, we consider a split-face approach, where a quadrilateral face is split into two triangles, and the third-order method for tetrahedra is directly applied. Also, we derive a second-derivative-free, high-order, volume quadrature formula for an arbitrary cell. Numerical results are presented for accuracy verification and applications with three-dimensional nontetrahedral grids.

Computational Fluid Dynamics↗