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NASA NTRS1976

Comparison of a two-dimensional shock impingement computation with experiment

Results of computations of two-dimensional viscous blunt-body flowfields with an impinging shock wave, with a time-dependent finite-difference method employed to solve the complete set of Navier-Stokes equations, are compared with experimental results. The experimental results were obtained in a 20-inch hypersonic tunnel with a planar shock impinging on the cylindrical leading edge of a fin, hence with the shock parallel to the centerline of the leading edge, so that type III and type IV interference patterns were generated. Close agreement is found. The overall effects of smoothing and grid size on the calculations are determined. A 31 x 51 mesh is adequate for wall pressure values (except in peaked regions).

Tannehill, J. C.↗

NASA NTRS1977

A Navier-Stokes solution of the three-dimensional viscous compressible flow in a centrifugal compressor impeller

A two-dimensional time-dependent computer code was utilized to calculate the three-dimensional steady flow within the impeller blading. The numerical method is an explicit time marching scheme in two spatial dimensions. Initially, an inviscid solution is generated on the hub blade-to-blade surface by the method of Katsanis and McNally (1973). Starting with the known inviscid solution, the viscous effects are calculated through iteration. The approach makes it possible to take into account principal impeller fluid-mechanical effects. It is pointed out that the second iterate provides a complete solution to the three-dimensional, compressible, Navier-Stokes equations for flow in a centrifugal impeller. The problems investigated are related to the study of a radial impeller and a backswept impeller.

Harp, J. L., Jr.↗

NASA NTRS1979

Numerical solution of compressible viscous flows

The presented review is concerned with the problem of calculating compressible viscous flows. Basic numerical considerations and problems associated with calculating viscous flows are examined and current numerical approaches toward the solution of the Navier-Stokes equations are discussed. It is pointed out that the numerical solution of the full time-dependent equations for turbulent flow is not practical with present computers. Therefore, turbulence effects must be accounted for by modeling. Developments related to turbulence modeling are described. In connection with a discussion of numerical methods for solving viscous flow equations, attention is given to numerical domains of dependence of typical explicit and implicit methods, the diffusion problem, the convection-diffusion problem, and the split-hybrid method.

Maccormack, R. W.↗

NASA NTRS1980

On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries

An inherent numerical problem associated with the fully explicit pseudospectral numerical simulation of the incompressible Navier-Stokes equation for viscous flows with no-slip walls is described. A semi-implicit scheme which circumvents this numerical difficulty is presented. In this algorithm the equation of continuity rather than the Poisson equation for pressure is solved directly. Pseudospectral formulation of the channel flow problem using Fourier series and Chebyshev polynomials expansions is given for this scheme. An example demonstrating the applicability of the method is given.

Moin, P.↗

NASA NTRS1975

Calculation of the flow on a blunted cone at high angle of attack

A new technique for calculating the entire flow-field on spherically blunted circular cones at high angles of attack and high Reynolds numbers is described. The calculations are based on a single-layer system of three-dimensional parabolic equations which are approximations to the full steady Navier-Stokes equations. Initial conditions at the sphere-cone tangency plane are provided by using an inviscid time-dependent solution added to a viscous nonsimilar boundary layer solution. Calculated results are compared with experimental heat transfer and pressure data for a 15 deg half-angle cone with a 1-in. spherical nose at 15 deg angle of attack. The free-stream Mach number is 10.6, and the free-stream Reynolds number is 1,200,000 per foot. Excellent agreement between the calculated and experimental data for both pressure and heat transfer is obtained.

Lubard, S. C.↗

NASA NTRS1974

An experimental and analytical study of slip and catalytic boundary conditions applied to spheres in low Reynolds number arc jet flows

Measurements of low-density dissociated arc-heated flows around low and high surface catalycity spheres are compared with numerical solutions to the full Navier-Stokes equations for reacting gas flow fields, taking into account slip and finite surface catalytic boundary conditions. A time-dependent finite difference solution technique developed by Li (1973, 1974) is used in the investigation and a simple transport model is employed for the flows considered. The correctness of the solution approach is demonstrated by the very good agreement obtained between experimental and computational data.

Scott, C. D.↗

NASA NTRS1981

Two-component Navier-Stokes computational model of viscous sublayer turbulence

A new computational method is presented for developing a quantitative model of viscous sublayer turbulence in incompressible flow. Appropriate space- and time-dependent boundary conditions are constructed for the three fluctuating velocity components at the outer edge of the viscous sublayer. This construction is formulated so as to model essential coherent structures observed in experiments. Time dependent Navier-Stokes equations are used to compute the characteristics of turbulence in the viscous sublayer.

Chapman, D. R.↗

NASA NTRS1982

A numerical solution of the supersonic flow over a rearward facing step with transverse non-reacting hydrogen injection

The present investigation represents an application of computational fluid dynamics to a problem associated with the flow in the combustor region of a supersonic combustion ramjet engine (scramjet). The governing equations are considered, taking into account the Navier-Stokes equations, a molecular viscosity calculation, the molecular thermal conductivity, molecular diffusion, and a turbulence model. The employed numerical solution is patterned after the explicit, time-dependent, unsplit, predictor-corrector, finite-difference method given by MacCormack (1969). The calculation is concerned with the supersonic flow over a rearward-facing step with transverse H2 injection at conditions germane to the combustor region of a scramjet engine. The H2 jet acts as an effective body which essentially shields the primary flow from the rearward-facing step, thus substantially changing the wave pattern in the primary flow.

Berman, H. A.↗

NASA NTRS1972

Convection in the tanks of a rotating spacecraft

A general study of convection and mixing of a stratified fluid in a rotating container is described, with application to the special problem of fluid heating and convection in a spacecraft tank. The analysis was based on a set of approximate equations for the Navier-Stokes description of fluid convection with small density variations in a rotating system, including effects of body forces due to temperature stratification (caused by a heater) and arbitrary time-dependent rotation of the tank about a noncentral axis. An efficient numerical finite difference scheme and computational method are described for the convection of vorticity and energy in a two-dimensional tank. Special procedures were developed for analysis of the thermodynamic states resulting from the approximate flow equations derived for small density variations. Results of the numerical simulation are presented for studying the effectiveness of rotation maneuvers in mixing stratified oxygen in the tanks of an Apollo spacecraft. Significant effects of the rotation maneuvers are discussed, including the reduction of the potential for pressure collapse.

Source record↗

NASA NTRS1973

Finite element analysis of unsteady incompressible flow around an oscillating obstacle of arbitrary shape.

An analytical procedure based on Navier-Stokes equations was developed for representing unsteady flow patterns around oscillating obstacles. A variational formulation of the Helmholtz vorticity equation was discretized in finite element form and integrated numerically. At each step of the numerical integration the velocity field around the obstacle was determined from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints at each time step of the numerical integration. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.

Bratanow, T.↗

NASA NTRS1976

Calculation of a separated turbulent boundary layer

The properties of a Navier-Stokes solution of a shock-separated turbulent flow over a flat wall are investigated. Refinements of an algebraic relaxation turbulence model previously shown to be of value for the simulation of separated flows are presented. A simplified analysis applicable near an adiabatic wall is developed and used to help verify the accuracy of the numerical solution. Features of the time-dependent response of a turbulent boundary layer to shock impingement are presented.

Baldwin, B.↗

NASA NTRS1978

Large eddy simulation of incompressible turbulent channel flow

The three-dimensional, time-dependent primitive equations of motion were numerically integrated for the case of turbulent channel flow. A partially implicit numerical method was developed. An important feature of this scheme is that the equation of continuity is solved directly. The residual field motions were simulated through an eddy viscosity model, while the large-scale field was obtained directly from the solution of the governing equations. An important portion of the initial velocity field was obtained from the solution of the linearized Navier-Stokes equations. The pseudospectral method was used for numerical differentiation in the horizontal directions, and second-order finite-difference schemes were used in the direction normal to the walls. The large eddy simulation technique is capable of reproducing some of the important features of wall-bounded turbulent flows. The resolvable portions of the root-mean square wall pressure fluctuations, pressure velocity-gradient correlations, and velocity pressure-gradient correlations are documented.

Moin, P.↗

DOE OSTI2025

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗

DOE OSTI2025

Optical neural engine for solving scientific partial differential equations

Abstract Solving partial differential equations (PDEs) is the cornerstone of scientific research and development. Data-driven machine learning (ML) approaches are emerging to accelerate time-consuming and computation-intensive numerical simulations of PDEs. Although optical systems offer high-throughput and energy-efficient ML hardware, their demonstration for solving PDEs is limited. Here, we present an optical neural engine (ONE) architecture combining diffractive optical neural networks for Fourier space processing and optical crossbar structures for real space processing to solve time-dependent and time-independent PDEs in diverse disciplines, including Darcy flow equation, the magnetostatic Poisson’s equation in demagnetization, the Navier-Stokes equation in incompressible fluid, Maxwell’s equations in nanophotonic metasurfaces, and coupled PDEs in a multiphysics system. We numerically and experimentally demonstrate the capability of the ONE architecture, which not only leverages the advantages of high-performance dual-space processing for outperforming traditional PDE solvers and being comparable with state-of-the-art ML models but also can be implemented using optical computing hardware with unique features of low-energy and highly parallel constant-time processing irrespective of model scales and real-time reconfigurability for tackling multiple tasks with the same architecture. The demonstrated architecture offers a versatile and powerful platform for large-scale scientific and engineering computations.

Tang, Yingheng (ORCID:0009000153622546)↗

NASA NTRS1974

Numerical calculations of velocity and pressure distribution around oscillating airfoils

An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.

Bratanow, T.↗

NASA NTRS1974

Analysis of three-dimensional unsteady flow around oscillating wings

A method based on the Navier-Stokes equations was developed for determining analytically the three-dimensional unsteady flow patterns around oscillating wings. The Helmholz vorticity transport equations were discretized in three-dimensional finite element form from a variational formulation and integrated numerically. At each time step of the numerical integration the velocity field was calculated from the representation of the three-dimensional wing by a system of optimized distribution of vortices in space. During the numerical integration of the vorticity transport equations the time-dependent boundary conditions on the wing were specified as external constraint conditions. Examples of obtained results describing the three-dimensional unsteady flow around a wing were presented.

Bratanow, T.↗

NASA NTRS1977

An experimental and computational investigation of the steady and unsteady transonic flow fields about an airfoil in a solid-wall test channel

An experimental and computational investigation of the steady and unsteady transonic flow field about a thick airfoil is described. An operational computer code for solving the two-dimensional, compressible Navier-Stokes equations for flow over airfoils was modified to include solid-wall, slipflow boundary conditions to properly assess the code and help guide the development of improved turbulence models. Steady and unsteady flow fields about an 18% thick circular arc airfoil at Mach numbers of 0.720, 0.754, and 0.783 and a chord Reynolds number of 11 x 10 to the 6th are predicted and compared with experiment. For the first time, computed results for unsteady turbulent flows with separation caused by a shock wave were obtained which qualitatively reproduce the time-dependent aspects of experiments. Features such as the intensity and reduced frequency of airfoil surface-pressure fluctuations, oscillatory regions of trailing-edge and shock-induced separation, and the Mach number range for unsteady flows were all qualitatively reproduced.

Levy, L. L., Jr.↗

NASA NTRS1981

VNAP2: A Computer Program for Computation of Two-dimensional, Time-dependent, Compressible, Turbulent Flow

A computer program, VNAP2, for calculating turbulent (as well as laminar and inviscid), steady, and unsteady flow is presented. It solves the two dimensional, time dependent, compressible Navier-Stokes equations. The turbulence is modeled with either an algebraic mixing length model, a one equation model, or the Jones-Launder two equation model. The geometry may be a single or a dual flowing stream. The interior grid points are computed using the unsplit MacCormack scheme. Two options to speed up the calculations for high Reynolds number flows are included. The boundary grid points are computed using a reference plane characteristic scheme with the viscous terms treated as source functions. An explicit artificial viscosity is included for shock computations. The fluid is assumed to be a perfect gas. The flow boundaries may be arbitrary curved solid walls, inflow/outflow boundaries, or free jet envelopes. Typical problems that can be solved concern nozzles, inlets, jet powered afterbodies, airfoils, and free jet expansions. The accuracy and efficiency of the program are shown by calculations of several inviscid and turbulent flows. The program and its use are described completely, and six sample cases and a code listing are included.

Cline, M. C.↗