A finite difference method for solving unsteady viscous flow problems
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Li, C. P.↗
Engineering topics
Li, C. P.
Publications and source records attributed to Li, C. P..
A 3-dimensional Navier-Stokes-Euler code for blunt-body flow computations
The shock-layer flowfield is obtained with or without viscous and heat-conducting dissipations from the conservative laws of fluid dynamics equations using a shock-fitting implicity finite-difference technique. The governing equations are cast in curvilinear-orthogonal coordinates and transformed to the domain between the shock and the body. Another set of equations is used for the singular coordinate axis, which, together with a cone generator away from the stagnation point, encloses the computation domain. A time-dependent alternating direction implicit factorization technique is applied to integrate the equations with local-time increment until a steady solution is reached. The shock location is updated after the flowfield computation, but the wall conditions are implemented into the implicit procedure. Innovative procedures are introduced to define the initial flowfield, to treat both perfect and equilibrium gases, to advance the solution on a coarse-to-fine grid sequence, and to start viscous flow computations from their corresponding inviscid solutions. The results are obtained from a grid no greater than 28 by 18 by 7 and converged within 300 integration steps. They are of sufficient accuracy to start parabolized Navier-Stokes or Euler calculations beyond the nose region, to compare with flight and wind-tunnel data, and to evaluate conceptual designs of reentry spacecraft.
Li, C. P.↗
Design study of an integrated aerobraking orbital transfer vehicle
An aerobraking orbital transfer vehicle (AOTV) concept, which has an aerobrake structure that is integrated with the propulsion stage, is discussed. The concept vehicle is to be assembled in space and is space-based. The advantages of aeroassist over an all propulsive vehicle are discussed and it is shown that the vehicle considered is very competitive with inflatable and deployable concepts from mass and performance aspects. The aerobrake geometry is an ellipsoidally blunted, raked-off, elliptical wide-angle cone with a toroidal skirt. Propellant tanks, engines, and subsystems are integrated into a closed, isogrid aerobrake structure which provides rigidity. The vehicle has two side-firing, gimbaled RL-10 type engines and carries 38,000 kg of useable propellant. The trajectory during aerobraking is determined from an adaptive guidance logic, and the heating is determined from engineering correlations as well as 3-D Navier-Stokes solutions. The AOTV is capable of placing 13,500 kg payload into geosynchronous Earth orbit (GEO) or carrying a LEO-GEO-LEO round-trip payload of 7100 kg. A two-stage version considered for lunar missions results in a lunar surface delivery capability of 18,000 kg or a round-trip capability of 6800 kg with 3860 kg delivery-only capability.
Scott, C. D.↗
A three-dimensional Navier-Stokes/Euler code for blunt-body flow computations
The formulation computation method of an improved version of the three-dimensional Navier-Stokes/Euler computation algorithm of Li (1981) for the numerical simulation of blunt-body reentry flows are discussed, and results for five sample problems are presented graphically. The vector notation of the coordinate systems is defined; the governing equations are presented in full; the Jacobian matrices and damping terms of the factorization technique (based on the alternating-direction implicit procedure of Beam and Warming, 1978) are explained; and the capabilities, limitations and proper use of the code are summarized. Examples presented include the equilibrium-flow problem for Shuttle-orbiter reentry at Mach 22 and angle of attack 40.8 deg and the perfect-gas problem of an aerobraking orbital-transfer vehicle with an ellipsoidal/60-deg cone and a toroidal sonic shoulder. The advantages of the improved code in terms of accuracy and computation time are indicated.
Li, C. P.↗
Numerical investigation of unsteady flow development in a nozzle-duct configuration
A computational approach has been used to resolve the transient phenomena caused by a rocket engine starting up and exhausting into a short duct. The results are obtained from the finite-difference Navier-Stokes, continuity, and energy equations for a single component gas. Using a recently developed numerical technique to account for the time-varying boundary conditions at the nozzle throat and the duct openings, the numerical simulation of the idealized ignition flow model has yielded some insight into the complex interactions between the wave patterns and the vortical flow. The present findings verify the hypothesis that a lateral expansion of the jetstream outside the nozzle will generate an over-pressure pulse on the duct wall if the jetstream is restricted within the nozzle by flow separation. The numerical simulations of this complex flow problem should be useful for improving simple correlation techniques used for engineering purposes.
Li, C. P.↗
On a finite-difference method for solving transient viscous flow problems
A method has been developed to solve the unsteady, compressible Navier-Stokes equation with the property of consistency and the ability of minimizing the equation stiffness. It relies on innovative extensions of the state-of-the-art finite-difference techniques and is composed of: (1) the upwind scheme for split-flux and the central scheme for conventional flux terms in the inviscid and viscous regions, respectively; (2) the characteristic treatment of both inviscid and viscous boundaries; (3) an ADI procedure compatible with interior and boundary points; and (4) a scalar matrix coefficient including viscous terms. The performance of this method is assessed with four sample problems; namely, a standing shock in the Laval duct, a shock reflected from the wall, the shock-induced boundary-layer separation, and a transient internal nozzle flow. The results from the present method, an existing hybrid block method, and a well-known two-step explicit method are compared and discussed. It is concluded that this method has an optimal trade-off between the solution accuracy and computational economy, and other desirable properties for analyzing transient viscous flow problems.
Li, C. P.↗
Space Shuttle orbiter entry heating and TPS response: STS-1 predictions and flight data
Aerothermodynamic development flight test data from the first orbital flight test of the Space Transportation System (STS) transmitted after entry blackout is given. Engineering predictions of boundary layer transition and numerical simulations of the orbiter flow field were confirmed. The data tended to substantiate preflight predictions of surface catalysis phenomena. The thermal response of the thermal protection system was as expected. The only exception is that internal free convection was found to be significant in limiting the peak temperature of the structure in areas which do not have internal insulation.
Ried, R. C.↗
Application of an implicit technique to the shock-layer flow around general bodies
The Euler and the parabolic Navier-Stokes equations are solved by an iterative implicit scheme along the body axis for angles of incidence up to 30 deg. A complete solution, including both the inviscid and viscous flows, can be obtained at stations normal to the axis. The technique is developed from a comparatively simple linearization procedure and has an option to iterate between body stations for higher accuracy. Reliable procedures have been introduced to account for the effect of the axial pressure gradient and to adjust the step increment for a given convergence requirement. A blunt-nosed conical and a Shuttle-Orbiter-like configuration were studied extensively and are compared with available laminar boundary-layer solutions and experimental data.
Li, C. P.↗
Numerical simulation of reentry flow around the Shuttle Orbiter including real gas effects
The flight environment of a reentry vehicle is predicted from the numerical solution of fluid dynamics equations for the region between the blunt nose and the swept wings. The inviscid portion of the shock layer is modeled by the Euler equations, but the laminar viscous flow adjacent to the wall is modeled by the approximate parabolic Navier-Stokes equations. The approximations made to the axial gradients of pressure and diffusive fluxes enable the coupled inviscid and viscous equations to be solved efficiently along the body axis. The equilibrium air aftbody code contains significant improvements over its predecessor which only considers a perfect gas model and noncircular configurations. The inclusion of numerical damping either explicitly or implicitly has extended its capabilities for predicting flow field around a winged configuration at higher Machs and greater angles of attack. The results are obtained on the cylindrical coordinates and satisfactory for the Shuttle Orbiter at a free-stream Mach number of 22 and an angle of attack of 40 deg. Also discussed are the inviscid formulation and its application for wind-tunnel conditions.
Li, C. P.↗
Computations of supersonic viscous flow over a finite-width plate
Finite difference methods were applied to solve the parabolic Navier-Stokes equations for the flow over a finite width plate at 0 deg and 10 deg angles of attack. The methods were developed on the basis of the operator factorization concept resulting in the split of a three dimensional equation into successive two dimensional equations. Backward and centered implicit factorization schemes, were used and their results were compared. Available numerical solutions and experimental data obtained at low Reynolds number conditions were also used for comparison. The backward implicit method provides a more successful solution, which ranges from the merged layer to the strong interaction regimes. Detailed structures were revealed of the shear layer around and behind the side edge.
Li, C. P.↗
Numerical computations of Orbiter flow fields and heating rates
Numerical computations of flow fields around an analytical description of the Space Shuttle Orbiter windward surface, including the root of the wing leading edge, are presented to illustrate the sensitivity of these calculations to several flow field modeling assumptions. Results of parametric flow field and boundary layer computations using the axisymmetric analogue concept to obtain three-dimensional heating rates, in conjunction with exact three-dimensional inviscid floe field solutions and two-dimensional boundary layer analysis - show the sensitivity of boundary layer edge conditions and heating rates to considerations of the inviscid flow field entropy layer, equilibrium air versus chemically and vibrationally frozen flow, and nonsimilar terms in the boundary layer computations. A cursory comparison between flow field predictions obtained from these methods and current Orbiter design methods has established a benchmark for selecting and adjusting these and future design methodologies.
Goodrich, W. D.↗
Recent developments on time-dependent calculation of nonequilibrium flows
It is shown that, in a time-dependent nonequilibrium flow analysis, erroneous and unstable solutions can result from the use of temperature equations and flow variables other than pressure in fitting the bow shock. A formulation is presented that has been critically evaluated by comparison with a previous one and found to yield steady and more accurate solutions in conserving total enthalpy.
Li, C. P.↗
Hypersonic nonequilibrium flow past a sphere at low Reynolds numbers
Detailed flow field properties are computed in the nose region of a sphere from the Navier-Stokes equations for a range of Reynolds numbers. Comparisons are made of theoretical flow fields with those obtained in experiments and also with other analyses. In particular, the effects of various approximations on the flow field are investigated. It is shown that the Navier-Stokes model consistently predicts a smaller region of disturbance upstream of a sphere than that in experiments.
Li, C. P.↗
Computations of non-reacting and reacting viscous blunt body flows, volume 1
The computer programs developed for computation of viscous shock layer flow distribution surrounding the nose of a shuttle orbiter during reentry are presented. The problem formulation and the numerical procedures used to solve the basic set of equations are described. The results of flow distribution properties at several trajectory points, ranging from the high altitude rarefied region to the low altitude boundary layer region are analyzed.
Li, C. P.↗
Computations of non-reacting and reacting viscous blunt body flows, volume 2
Computer programs for calculating the flow distribution in the nose region of a blunt body at arbitrary speed and altitude are discussed. The programs differ from each other in their ability to consider either thin shock or thick shock conditions and in the use of either ideal, equilibrium air, or nonequilibrium air chemistry. The application of the programs to analyzing the flow distribution around the nose of the shuttle orbiter during reentry is reported.
Li, C. P.↗
Numerical solution of viscous reacting blunt body flows of a multicomponent mixture.
Hypersonic flow of a chemically nonequilibrium multicomponent mixture past an axisymmetric blunt body is considered on the basis of the exact Navier-Stokes and chemical rate equations. The flowfield, characterized by a sharp shock and a boundary layer, is solved numerically in the region between the shock and the body. The governing equations are first formulated in generalized orthogonal coordinates, and then recast in computational coordinates in which both the shock and the body are two straight lines and one coordinate is stretched to allow higher space resolution near the body. The thermodynamic and transport properties are obtained from the most rigorous theories and reliable data for each component and for the mixture itself. A time-marching second-order finite-difference method is used to solve the equations.
Li, C. P.↗
A viscous starter solution for shuttle flow field computations
A time dependent numerical procedure that calculates the viscous flow in the shock layer around the shuttle nose has been developed. It has been applied to the free stream conditions that may be encountered in the shuttle flight trajectory and found to be efficient and in satisfactory agreement with other steady and unsteady techniques. Two features associated with the present formulation are responsible for its successful applications. The first is that the flow field computation is made within the shock layer; the second is that the mapping of the computational region places proportionately more mesh points in the vicinity of the wall than in the remaining computational region. The numerical results may be used for low altitude flight to determine the edge conditions for a thin boundary layer, to provide initial boundary layer profiles for downstream boundary layer calculations, and to start supersonic computations. For high altitude flight, it may be used to determine the complete flow field in the nose region and also provide a starter solution for subsequent supersonic flow computations.
Li, C. P.↗
Time-dependent solutions of nonequilibrium dissociating flow past a blunt body
Nonequilibrium dissociating gas flow past blunt body using time dependent shock layer analysis
Li, C. P.↗