On axioms for heterogeneous continua.
Mathematical model for heterogeneous continua, discussing kinematical definitions, conservation principles, thermodynamics
THERMODYNAMICS↗
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Results for “flow conservation”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Numerical solution of nonlinear partial differential equations of mixed type
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
Jameson, A.↗
An empirical model for the solar wind velocity
An analytic expression for the average radial component of the Solar Wind velocity between 1 solar radius and 1 AU is developed. The model is constructed by assuming the conservation of particle flow in the Solar Wind and application of a twelve-year average measured value of the Solar Wind radial velocity at 1 AU.
Berman, A. L.↗
TF34 engine compression system computer study
The stability of the fan and the compressor components was examined individually using linearized and time dependent, one dimensional stability analysis techniques. The stability of the fan core integrated compression system was investigated using a two dimensional compression system model. The analytical equations on which this model was based satisfied the mass, axial momentum, radial momentum, and energy conservation equations for flow through a finite control volume. The results gave an accurate simulation of the flow through the compression system. The speed lines of the components were reproduced; the points of instability were accurately predicted; the locations where the instability was initiated in the fan and the core were indicated; and the variation of the bypass ratio during flow throttling was calculated. The validity of the analytical techniques was then established by comparing these results with test data and with results obtained from the steady state cycle deck.
Hosny, W. M.↗
Applications of the concept of generalized vorticity to space plasmas
A reformulation of the momentum equation for electrons or ions in a collisionless plasma leads to an equation which describes the behavior of the plasma in terms of a generalized vorticity. This vorticity is both divergence-free and conserved along plasma flow streamlines. When the plasma has zero vorticity, a special relation is established which appears to have application to small scale magnetic features within both conventional space plasmas and superconductors.
Banks, P. M.↗
Experiments on a round turbulent buoyant plume
This paper reports a comprehensive set of hot-wire measurements of a round buoyant plume which was generated by forcing a jet of hot air vertically up into quiescent environment. The boundary conditions of the experiment were measured, and are documented in the present paper in an attempt to sort out the contradictory mean flow results from the earlier studies. The ambient temperature was monitored to insure that the facility was not stratified and that the experiment was conducted in a neutral environment. The axisymmetry of the flow was checked by using a planar array of sixteen thermocouples and the mean temperature measurements from these are used to supplement the hot-wire measurements. The source flow conditions were measured so as to ascertain the rate at which the buoyancy was added to the flow. The measurements conserve buoyancy within 10 percent. The results are used to carry out the balances of the mean energy and momentum differential equations. In the mean energy equation it is found that the vertical advection of the energy is primarily balanced by the radial turbulent transport. In the mean momentum equation the vertical advection of momentum and the buoyancy force balance the radial turbulent transport. The buoyancy force is the second largest term in this balance and is responsible for the wider (and higher) velocity profiles in plumes as compared to jets. Budgets of the temperature variance and turbulence kinetic energy are also carried out in which thermal and mechanical dissipation rates are obtained as the closing terms. Similarities and differences between the two balances are discussed. It is found that even though the direct affect of buoyancy on turbulence, as evidenced by the buoyancy production term, is substantial, most of the turbulence is produced by shear. This is in contrast to the mean velocity field where the affect of buoyancy force is quite strong. Therefore, it is concluded that in a buoyant plume the primary affect of buoyancy on turbulence is indirect, and enters through the mean velocity field (giving larger shear production).
Shabbir, Aamir↗
Calculation of Propulsive Nozzle Flowfields in Multidiffusing Chemically Reacting Environments
An advanced engineering model has been developed to aid in the analysis and design of hydrogen/oxygen chemical rocket engines. The complete multispecies, chemically reacting and multidiffusing Navier-Stokes equations are modelled, including the Soret thermal diffusion and the Dufour energy transfer terms. In addition to the spectrum of multispecies aspects developed, the model developed in this study is also conservative in axisymmetric flow for both inviscid and viscous flow environments and the boundary conditions employ a viscous, chemically reacting, reference plane characteristics method. Demonstration cases are presented for a 1030:1 area ratio nozzle, a 25 lbf film cooled nozzle, and a transpiration cooled plug and spool rocket engine. The results indicate that the thrust coefficient predictions of the 1030:1 and the 25 lbf film cooled nozzle are within 0.2 to 0.5 percent, respectively, of experimental measurements when all of the chemical reaction and diffusion terms are considered. Further, the model's predictions agree very well with the heat transfer measurements made in all of the nozzle test cases. The Soret thermal diffusion term is demonstrated to have a significant effect on the predicted mass fraction of hydrogen along the wall of the nozzle in both the laminar flow 1030:1 nozzle and the turbulent flow plug and spool nozzle analysis cases performed. Further, the Soret term was shown to represent an important fraction of the diffusion fluxes occurring in a transpiration cooled rocket engine.
Kacynski, Kenneth John↗
Numerical simulations of small amplitude acoustic wave propagation in a converging-diverging nozzle
Numerical simulations of the small amplitude acoustic wave propagation in a converging-diverging nozzle proposed in Category 5 are performed and presented here. The quasi 1-D unsteady flow equations in conservative form are discretized by the DRP scheme with the artificial damping developed by Tam and Webb. Characteristic boundary conditions are used at both the inlet and outlet of the nozzle. The effect of different numerical implementations of the subsonic inflow boundary conditions on the convergence of the solution to the steady state is studied. In the case of a subsonic outflow in which a shock is formed in the nozzle, the interaction between acoustics and the shock is also investigated.
Dong, Thomas Z.↗
Some Effects of Compressibility on the Flow Through Fans and Turbines
The laws of conservation of mass, momentum, and energy are applied to the compressible flow through a two-dimensional cascade of airfoils. A fundamental relation between the ultimate upstream and downstream flow angles, the inlet Mach number, and the pressure ratio across the cascade is derived. Comparison with the corresponding relation for incompressible flow shows large differences. The fundamental relation reveals two ranges of flow angles and inlet Mach numbers, for which no ideal pressure ratio exists. One of these nonideal operating ranges is analogous to a similar type in incompressible flow. The other is characteristic only of compressible flow. The effect of variable axial-flow area is treated. Some implications of the basic conservation laws in the case of nonideal flow through cascades are discussed.
Perl, W.↗
A data-driven sensor placement approach for detecting voltage violations in distribution systems
Stochastic fluctuations in power injections from distributed energy resources (DERs) combined with load variability can cause constraint violations (e.g., exceeded voltage limits) in electric distribution systems. To monitor grid operations, sensors are placed to measure important quantities such as the voltage magnitudes. Here, in this paper, we consider a sensor placement problem which seeks to identify locations for installing sensors that can capture all possible violations of voltage magnitude limits. We formulate a bilevel optimization problem that minimizes the number of sensors and avoids false sensor alarms in the upper level while ensuring detection of any voltage violations in the lower level. This problem is challenging due to the nonlinearity of the power flow equations and the presence of binary variables. Accordingly, we employ recently developed conservative linear approximations of the power flow equations that overestimate or underestimate the voltage magnitudes. By replacing the nonlinear power flow equations with conservative linear approximations, we can ensure that the resulting sensor locations and thresholds are sufficient to identify any constraint violations. Additionally, we apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors. Lastly, we improve the quality of the results via an approximate gradient descent method that adjusts the sensor thresholds. We demonstrate the effectiveness of our proposed method for several test cases, including a system with multiple switching configurations.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Numerical Modeling of Conjugate Heat Transfer in Fluid Network
Fluid network modeling with conjugate heat transfer has many applications in Aerospace engineering. In modeling unsteady flow with heat transfer, it is important to know the variation of wall temperature in time and space to calculate heat transfer between solid to fluid. Since wall temperature is a function of flow, a coupled analysis of temperature of solid and fluid is necessary. In cryogenic applications, modeling of conjugate heat transfer is of great importance to correctly predict boil-off rate in propellant tanks and chill down of transfer lines. In TFAWS 2003, the present author delivered a paper to describe a general-purpose computer program, GFSSP (Generalized Fluid System Simulation Program). GFSSP calculates flow distribution in complex flow circuit for compressible/incompressible, with or without heat transfer or phase change in all real fluids or mixtures. The flow circuit constitutes of fluid nodes and branches. The mass, energy and specie conservation equations are solved at the nodes where as momentum conservation equations are solved at the branches. The proposed paper describes the extension of GFSSP to model conjugate heat transfer. The network also includes solid nodes and conductors in addition to fluid nodes and branches. The energy conservation equations for solid nodes solves to determine the temperatures of the solid nodes simultaneously with all conservation equations governing fluid flow. The numerical scheme accounts for conduction, convection and radiation heat transfer. The paper will also describe the applications of the code to predict chill down of cryogenic transfer line and boil-off rate of cryogenic propellant storage tank.
Majumdar, Alok↗
Unsteady delta-wing flow computation using an implicit factored Euler scheme
The conservative unsteady Euler equations for the flow relative motion in the moving frame of reference are used to solve for the steady and unsteady flows around sharp-edged delta wings. The resulting equations are solved by using an implicit approximately-factored finite-volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in cross-flow planes at the grid chord stations. The computational applications cover a steady flow around a delta wing whose results serve as the initial conditions for the unsteady flow around a pitching delta wing about a large angle of attack. The steady results are compared with the experimental data and the periodic solution is achieved within the third cycle of oscillation.
Kandil, Osama A.↗
Computation of vortex-dominated flow for a delta wing undergoing pitching oscillation
The conservative, unsteady Euler equations for the flow relative to a moving frame of reference are used to solve for the three-dimensional steady and unsteady flows around a sharp-edged delta wing. The resulting equations are solved by using an implicit, approximately factored, finite-volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in crossflow planes at the grid-chord stations. The computational applications cover a steady flow around a delta wing, whose results serve as the initial conditions for the unsteady flow around a pitching delta wing at a large mean angle of attack. The steady results are compared with the experimental data, and the unsteady results are compared with results of a flux-difference splitting scheme.
Kandil, Osama A.↗
Thermocapillary Flow and Aggregation of Bubbles on a Solid Wall
During the electrolytic evolution of oxygen bubbles forming on a vertically oriented transparent tin oxide electrode, bubbles were found to be mutually attractive. The mechanism of the aggregation had never been explained satisfactorily until Guelcher et al. attributed it to thermocapillary flow. The gradient of surface tension of the liquid at the bubble's surface, which was established because of reaction heat and ohmic heat loss at the electrode wall, drives flow of the liquid adjacent to each bubble; the bubble "pumps" fluid along its surface away from the wall. Fluid flows toward the bubble to conserve mass and entrains nearby bubbles in the flow pattern. The same logic would apply when two bubbles of equal size are adjacent to each other on a warm wall. Each bubble drives thermocapillary flow and hence entrains the other in its flow pattern, which drives the aggregation. Our objective here is to perform experiments where the temperature gradient at the wall is well known and controlled. The theory can be quantitatively tested by studying aggregation of bubble pairs of equal size, and by varying system parameters such as temperature gradient, bubble size and fluid viscosity. The results are then compared with the theory in a quantitatively rigorous manner. We demonstrate that the theory without adjustable parameters is capable of quantitatively modeling the rate of aggregation of two bubbles. The equations governing the thermocapillary flow around a single stationary bubble on a heated or cooled wall in a semi-infinite domain were solved. Both Reynolds number and Marangoni number were much less than unity. The critical result is that liquid in the vicinity of a warm wall flows toward a stationary collector bubble. Consequently the thermocapillary flow around the stationary bubble entrains another bubble toward itself. The bubbles undergo hindered translation parallel to the wall with velocity U while the fluid flow field is described with u. Two velocities were equated by using a wall hindrance parameter q: U = qu [1] which shows the velocity of bubble is proportional to the entraining velocity. The hindrance parameter q can experimentally be measured independently. q can also be calculated by solving the equations of motion for a bubble translating parallel to a solid wall. The experimental cell is cylindrical with an ID of 10 cm and consists of a 1 cm deep main cell filled with silicone oil and flanked by two thermal reservoirs. The upper thermal reservoir was heated and the lower thermal reservoir was cooled so that the bubbles aggregate. Two types of silicone oil (eta = 0.02 and 0.50 Pa s) were used. Two equal sized air bubbles were injected into the cell with a syringe. The center-to-center distance of bubbles was observed through a microscope. Bubble radius ranged from 0.40 mm to 0.65 mm and the temperature gradients along with the cell ranged from 1400 to 5000 K/m. The bubbles aggregated when heat flows from the wall to the fluid. The velocities of bubbles were in the range of 1 - 10 microns/s. The separation r decreased more quickly when the temperature gradient was higher, bubble size was larger, and the oil viscosity was lower. r decreased more rapidly as the bubbles approached each other. Dimensionless time was arbitrarily set to be zero when the dimensionless center-to-center distance between the bubbles was 4. All the bubble trajectories fall onto one line, especially in the range of dimensionless distance from 4 to 3. This means the relative movement of the bubble pair is proportional to the temperature gradient and bubble size and it is inversely proportional to the viscosity of the oil. This result strongly suggests that the thermocapillary flow-based aggregation mechanism is correct. A value of q can be estimated by fitting the scaled data to Eq. [1]. A best fit value of q was obtained as q = 0.26 with a standard deviation of 0.03. Independent experimental results for q for a 0.5 mm radius bubble, give values of q in the range 0.11 to 0.23. The value of q obtained from solving the equations of motion reveals q has values in the range 0.23 - 0.30. Since the full scale of possible values of q is zero to one, the maximum deviation of independently determined values of q from the best fit value was 15% of this full scale. Thus reasonable quantitative agreement between theory and experiment has been obtained.
Kasumi, Hiroki↗
Thickening oscillation of a delta wing using Navier-Stokes and Navier-displacement equations
The problem of unsteady, supersonic, locally conical, vortical flow around a delta wing undergoing thickening oscillation is solved using the unsteady, thin-layer Navier-Stokes equations and the unsteady linearized Navier-displacement equations. The unsteady, thin-layer Navier-Stokes equations are solved using the implicit approximate-factorization finite-volume scheme to compute the conservative components of the flow vector field. With the conservative components known at any time step, the linearized, Navier-displacement equations are solved using the alternating, direction-implicit scheme to obtain the grid points displacements due to known displacement boundary conditions. A grid-displacements limiter, in the form of a low mesh Reynolds number, is used to limit grid-folding in regions of highly reversed flow.
Chuang, Hsin-Kung A.↗
The method of space-time conservation element and solution element-applications to one-dimensional and two-dimensional time-marching flow problems
A nontraditional numerical method for solving conservation laws is being developed. The new method is designed from a physicist's perspective, i.e., its development is based more on physics than numerics. Even though it uses only the simplest approximation techniques, a 2D time-marching Euler solver developed recently using the new method is capable of generating nearly perfect solutions for a 2D shock reflection problem used by Helen Yee and others. Moreover, a recent application of this solver to computational aeroacoustics (CAA) problems reveals that: (1) accuracy of its results is comparable to that of a 6th order compact difference scheme even though nominally the current solver is only of 2nd-order accuracy; (2) generally, the non-reflecting boundary condition can be implemented in a simple way without involving characteristic variables; and (3) most importantly, the current solver is capable of handling both continuous and discontinuous flows very well and thus provides a unique numerical tool for solving those flow problems where the interactions between sound waves and shocks are important, such as the noise field around a supersonic over- or under-expansion jet.
Chang, Sin-Chung↗
A generalized hyperbolic marching technique for three-dimensional supersonic flow with shocks
The current paper introduces a generalized, nonorthogonal coordinate system for application to supersonic flows with such large angles between streamlines of the initial Cauchy data and the marching direction that the velocity component in the marching direction is actually subsonic. Finite-difference operators are developed to solve the three-dimensional, steady, inviscid conservation equations of fluid flow referenced to this general frame. Moreover, a new method of aligning the difference mesh to the bow shock wave is presented which eliminates both the necessity for differencing the free-stream flow properties and the spurious fluctuations that often arise in conservative variables when they are differenced across the discontinuity. Results are given for the case of flow past a conical surface.
Rizzi, A. W.↗
Applications of the Method of Space-Time Conservation Element and the Solution Element to Unsteady Chemically Reactive Flows
This document reports the conclusion and findings of our research activities for this grant. The goal of the project is the development and application of the method of Space-Time Conservation Element and Solution Element, or the CE/SE method, to simulate chemically reacting flows. The product of this project will be a high-fidelity, time-accurate flow solver analyzing unsteady flow fields advanced propulsion concepts, including the low-emission turbojet engine combustion and flow fields of the Pulse Detonation Engines (PDE). Based on the documents and computer software of the CE/SE method that we have received from the CE/SE working group at NASA Lewis, we have focused our research effort on addressing outstanding technical issues related to the extension of the CE/SE method for unsteady, chemically reactive flows. In particular, we have made progresses in the following three aspects: (1) Derivation of the governing equations for reacting flows; (2) Numerical treatments of stiff source terms; and (3) Detailed simulations of ZND detonation waves.
Yu, Sheng-Tao↗