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NASA NTRS1984

Navier-Stokes calculations for unsteady three-dimensional vortical flows in unbounded domains

Finite-difference Navier-Stokes calculations for unsteady, three-dimensional, incompressible, viscous flows induced by initial vorticity distributions are presented and discussed in this paper. The initial vorticity distributions are assumed to be embedded in a flow field of infinite extent that is quiescent at infinity. These vorticity distributions are typical of vortex rings and other closed vortical tubes or structures. Such structures are important elements in fluid flows such as jets, atmospheric convection and the far-field wakes of aircraft; studies of their interaction may aid in an understanding of complex fluid flows. The calculations employ a method recently proposed by Ting to approximate the infinite-domain boundary value problem with a finite boundary computational domain, and this method is shown to yield accurate three-dimensional results for reasonable expenditures of computer time. Because of the efficiency of the boundary condition technique and the resulting Navier-Stokes code, a 16-bit minicomputer with virtual memory was capable of performing the calculations for the unsteady motion of two obliquely colliding vortex rings. The results of these calculations are presented in the paper.

Chamberlain, J. P.↗

NASA NTRS1984

Dynamic response of shock waves in transonic diffuser and supersonic inlet - An analysis with the Navier-Stokes equations and adaptive grid

An existing method which solves the multi-dimensional ensemble-averaged compressible time-dependent Navier-Stokes equations in conjunction with mixing length turbulence model and shock capturing technique has been extended to include the shock-tracking adaptive grid systems. The numerical scheme for solving the governing equations is based on a linearized block implicit approach. The effects of grid-motion and grid-distribution on the calculated flow solutions have been studied in relative detail and this is carried out in the context of physically steady, shocked flows computed with non-stationary grids. Subsequently, the unsteady dynamics of the flows occurring in a supercritically operated transonic diffuser and a mixed compression supersonic inlet have been investigated with the adaptive grid systems by solving the Navier-Stokes equations.

Liu, N.-S.↗

NASA NTRS1984

Three-dimensional Navier-Stokes calculations of multiple interacting vortex rings

Results from a finite-difference Navier-Stokes code for three-dimensional, unsteady, vortical flows in unbounded domains are presented and analyzed in this paper. The vortical flows presented are representative of vortex rings and other closed vortical tubes or structures in fluid mechanics. Such structures are important elements in fluid flows such as jets, atmospheric turbulence, and the far-field wakes of aircraft, and studies of their interaction may aid in an understanding of complex fluid flows. The paper demonstrates that computational methods can be used as a viable alternative or supplement to experimental techniques for studying the physics of vortex flows. The separate visualization of vortex stretching, convection, and diffusion is presented in this paper for a single elliptical vortex ring.The calculations employ a truncated series expansion technique to simulate the unbounded nature of the fluid flow with a finite computational domain, which is a more accurate technique than the conventional freestream boundary specification. The numerical divergence of the three-dimensional vorticity field is considered as a useful estimate of truncation error, and the use of a kinetic energy decay law as a calculation check is demonstrated. Results from the Navier-Stokes code are presented for the unsteady motion of two and four vortex rings along parallel axes, and the results agree qualitatively with experimental flow visualization.

Chamberlain, J. P.↗

NASA NTRS1982

Comparison of solutions of the Navier-Stokes and interacting boundary layer equations for separated turbulent flow

Comparisons are presented between numerical solutions of the Navier-Stokes and interacting boundary-layer equations for the subsonic separated turbulent flow over an axisymmetric boattail configuration for which experimental data was available. Comparisons were made using a standard algebraic turbulence model and its modification using a streamwise relaxation model. Grid size studies were performed to provide an approximate assessment of the sensitivity of each method to mesh size. In general, good agreement was observed between the solutions of the Navier-Stokes and interacting boundary-layer equations; however, the interaction approach responded more strongly to the modified turbulence model thereby giving better agreement with experimental data.

Vatsa, V. N.↗

NASA NTRS1984

A note on the variational method of Stokes and DeMarcus for radiative transfer in planetary atmospheres

A generalization of the variational principle of Stokes and DeMarcus (1971) by the construction of two bifunctionals is proposed. The equation of radiative transfer is obtained by imposing the stationary condition on such functionals, and the reflectivity and transmissivity at arbitrary angle are given by their extremum values. Further simplification is possible when trail functions are used that are linear in the variational parameters, in which case the source function derived from the method of Stokes and DeMarcus can give the reflectivity at arbitrary emergent angle to second order. The usefulness and limitations of these variational priinciples are explored with extensive numerical results.

Chow, K. S. K.↗

NASA NTRS1985

Hydrodynamic stability problem formulated by numerical solutions of the Navier-Stokes equations

The problem of hydrodynamic stability and the transition from laminar to turbulent flows are reformulated by seeking numerical solutions of the full, unapproximated Navier-Stokes equations. This method differs significantly from the well known Orr-Sommerfeld equation approach. The oncoming laminar flow is disturbed by forced, time dependent perturbations. The magnitudes of these perturbations are arbitrary. Then, the ensuing spatial and temporal development of the imposed perturbations on the basic flow is calculated by direct numerical solutions of the time dependent Navier-Stokes equations. One of the main advantages of this method is its ability to simulate nonlinear processes. As a specific application of this technique to SSME (Space Shuttle Main Engine) flow configurations, computer programs have been written for the two dimensional flow over a backward facing step. This numerical code will be tested for operational use as part of a continued research collaboration effort with the NASA/MSFC counterparts.

Hyun, J. M.↗

NASA NTRS1985

The San Fernando Observatory video Stokes polarimeter

A study was conducted to determine the suitability of the San Fernando Observatory's 61 cm (24 inch) aperture vacuum solar telescope and 3 m (118 inch) focal length vacuum spectroheliograph for Stokes Polarimetry measurements. The polarization characteristics of these two instruments was measured by determining their Mueller matrices as a function of telescope orientation, field angle, wavelength, grating type, and position of the measuring beam in the telescope entrance window. In general, the polarizing and depolarizing properties are small so that inversion of the system Mueller matrix will permit the accurate measurement of Stokes profiles for vector magnetic field determination. A proposed polarimeter design based on the use of a TV camera system to simultaneously scan six different polarization components of a given line profile is described. This design, which uses no rotating optics or electronic modulators and makes efficient use of the available irradiance, promises to yield high quality vector magnetograms.

Richter, P. H.↗

NASA NTRS1985

Least squares inversion of Stokes profiles in the presence of velocity gradients

The Auer, Heasley and House Stokes inversion procedure in use at High Altitude Observatory is based on the analytic solution of the equation of transfer for polarized light where the representation of the thermodynamic and magnetic structure of the atmosphere is assumed to have a high degree of invariance, namely, a Milne-Eddington (ME) structure with a constant magnetic field. In the presence of invariance breaking gradients the resultant Stokes profiles are represented only approximately, if at all, by analytic forms. The accuracy of the inversion parameters and their significance as measures of actual structure are explored for the ME and the Landman-Finn sunspot models under the effects of velocity gradients. The resulting field parameters are good to a few percent and prove to be insensitive to the errors committed by the use of a ME-representation, but the resulting ME parameters yield a less precise measure of thermal structure.

Skumanich, A.↗

NASA NTRS1985

Time-dependent Navier-Stokes computations of separated flows over airfoils

The application of both a central-difference and an upwind implicit approximate factorization Navier-Stokes algorithm to highly separated flow is described. Using the thin-layer approximation, both algorithms are employed to solve the low Reynolds number laminar flow around a circular cylinder with periodic shedding. Results agree with experimental Strouhal numbers to within about 10 percent. A grid refinement study is conducted using the upwind code for both the thin-layer and complete Navier-Stokes approximations. The circular cylinder solution is shown to be highly dependent upon both grid density outward from the body and grid extent. Finally, the thin-layer central-difference code is used to predict the time-accurate unsteady flow about two stalled airfoils. The NACA 0012 airfoil turbulent solutions are periodic with Strouhal numbers of about 0.1; a strong vortex is periodically shed from the leading edge. The LS(1)-0417 airfoil turbulent solution shows a high frequency oscillatory trailing edge separation. The 20 degree NACA 0012 airfoil solution is massively separated, but shows no evidence of periodicity.

Rumsey, C. L.↗

NASA NTRS1985

A parabolized Navier-Stokes algorithm for separated supersonic internal flows

A stable global-iteration procedure is applied to a fully implicit noniterative parabolized Navier-Stokes algorithm to improve the solutions of the single-sweep method as well as to resolve small separation bubbles due to shock-wave/boundary-layer interactions. A modified form of the FLARE approximation is employed in the separated regions and some requirements for stable iterations are discussed. The results of the global-iteration scheme for two test cases of supersonic internal flows are compared against the results of the single-sweep method as well as the results of the full Navier-Stokes solver.

Chitsomboon, T.↗

NASA NTRS1985

Adapting a Navier-Stokes code to the ICL-DAP

The results of an experiment are reported, i.c., to adapt a Navier-Stokes code, originally developed on a serial computer, to concurrent processing on the CL Distributed Array Processor (DAP). The algorithm used in solving the Navier-Stokes equations is briefly described. The architecture of the DAP and DAP FORTRAN are also described. The modifications of the algorithm so as to fit the DAP are given and discussed. Finally, performance results are given and conclusions are drawn.

Grosch, C. E.↗

NASA NTRS1986

Transonic Navier-Stokes wing solution using a zonal approach. Part 1: Solution methodology and code validation

A fast diagonalized Beam-Warming algorithm is coupled with a zonal approach to solve the three-dimensional Euler/Navier-Stokes equations. The computer code, called Transonic Navier-Stokes (TNS), uses a total of four zones for wing configurations (or can be extended to complete aircraft configurations by adding zones). In the inner blocks near the wing surface, the thin-layer Navier-Stokes equations are solved, while in the outer two blocks the Euler equations are solved. The diagonal algorithm yields a speedup of as much as a factor of 40 over the original algorithm/zonal method code. The TNS code, in addition, has the capability to model wind tunnel walls. Transonic viscous solutions are obtained on a 150,000-point mesh for a NACA 0012 wing. A three-order-of-magnitude drop in the L2-norm of the residual requires approximately 500 iterations, which takes about 45 min of CPU time on a Cray-XMP processor. Simulations are also conducted for a different geometrical wing called WING C. All cases show good agreement with experimental data.

Flores, J.↗

NASA NTRS1986

Pseudo-time algorithms for the Navier-Stokes equations

A pseudo-time method is introduced to integrate the compressible Navier-Stokes equations to a steady state. This method is a generalization of a method used by Crocco and also by Allen and Cheng. We show that for a simple heat equation that this is just a renormalization of the time. For a convection-diffusion equation the renormalization is dependent only on the viscous terms. We implement the method for the Navier-Stokes equations using a Runge-Kutta type algorithm. This permits the time step to be chosen based on the inviscid model only. We also discuss the use of residual smoothing when viscous terms are present.

Swanson, R. C.↗

NASA NTRS1986

Navier-Stokes computations for circulation controlled airfoils

Navier-Stokes computations of subsonic to transonic flow past airfoils with augmented lift due to rearward jet blowing over a curved trailing edge are presented. The approach uses a spiral grid topology. Solutions are obtained using a Navier-Stokes code which employs an implicit finite difference method, an algebraic turbulence model, and developments which improve stability, convergence, and accuracy. Results are compared against experiments for no jet blowing and moderate jet pressures and demonstrate the capability to compute these complicated flows.

Pulliam, T. H.↗

NASA NTRS1986

Transonic Navier-Stokes wing solutions using a zonal approach. Part 2: High angle-of-attack simulation

A computer code is under development whereby the thin-layer Reynolds-averaged Navier-Stokes equations are to be applied to realistic fighter-aircraft configurations. This transonic Navier-Stokes code (TNS) utilizes a zonal approach in order to treat complex geometries and satisfy in-core computer memory constraints. The zonal approach has been applied to isolated wing geometries in order to facilitate code development. Part 1 of this paper addresses the TNS finite-difference algorithm, zonal methodology, and code validation with experimental data. Part 2 of this paper addresses some numerical issues such as code robustness, efficiency, and accuracy at high angles of attack. Special free-stream-preserving metrics proved an effective way to treat H-mesh singularities over a large range of severe flow conditions, including strong leading-edge flow gradients, massive shock-induced separation, and stall. Furthermore, lift and drag coefficients have been computed for a wing up through CLmax. Numerical oil flow patterns and particle trajectories are presented both for subcritical and transonic flow. These flow simulations are rich with complex separated flow physics and demonstrate the efficiency and robustness of the zonal approach.

Chaderjian, N. M.↗

NASA NTRS1985

The Haleakala Stokes polarimeter

A versatile Stokes polarimeter for solar observations has been developed. Recent improvements to the instrument include a high-resolution echelle spectrometer coupled to the telescope by optical fibers, and 128-element diode array detectors. The on-axis design on the telescope and polarimeter limit instrumental polarization to .0001, and the spectrometer detector combination provides spectral resolving power of 160,000 for any wavelength between 4000 and 11,000 A. This paper describes the Haleakala polarimeter and in particular the spectrometer with its fiber-optic coupling. Examples of Stokes line profiles observed in a sunspot are presented, together with derived vector magnetic field maps.

Mickey, D. L.↗

NASA NTRS1986

Upwind Navier-Stokes solutions for separated periodic flows

The application of an upwind implicit approximate factorization Navier-Stokes algorithm to highly separated flow is described. Using both the thin-layer and complete forms of the Navier-Stokes equations, the low Reynolds number laminar flow around a circular cylinder with periodic shedding is solved. The effect of grid density, grid extent, and time step on the Strouhal number is shown. Results from both sets of equations agree within the experimental data band. Unsteady, laminar flow computations around inclined plates and separated airfoils are also described. Strouhal numbers agree to within 5 percent of experiments for inclined plates. Differences between the complete equations and the thin-layer approximation for separated periodic flows are discussed. Computations of an impulsively started circular cylinder and airfoil yield time-accurate flowfield shapes in good agreement with experimental flow visualizations. The turbulent computation of an airfoil at a high angle-of-attack is massively separated, but shows no evidence of periodicity.

Rumsey, C. L.↗

NASA NTRS1986

SSME thrust chamber modeling with Navier Stokes equations

The capability of predicting two-dimensional, compressible and reacting flow in the combustion chamber and nozzle of the Space Shuttle Main Engine (SSME) is demonstrated. A nonorthogonal body fitted coordinate system has been used to represent the combustor and nozzle geometry. The Navier-Stokes equations are solved for the entire thrust chamber with the k-epsilon turbulence model accounting for compressibility and large pressure gradients effects. Results of the computational test cases reveal all expected features of the transonic nozzle flows including location of sonic line, internal shock and boundary layer build-up. Calculated performance parameters such as thrust, flow rate, and specific impulse are also in reasonble agreement with available data. The results show promising potential of solving full Navier-Stokes equations with heat transfer and two-phase combustion in truly comprehensive modeling of rocket engines.

Przekwas, A. J.↗