Net Mass Flow Components in a Three-Dimensional Unsteady Rotor Inflow Model

2003 ◽  
Author(s):  
Ke Yu ◽  
David A. Peters
Keyword(s):  
Mass Flow ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Skew Angle ◽  
Pressure Potential ◽  
Associated Legendre Functions ◽  
Galerkin Approach ◽  
Flow Equations ◽  
Closed Form Solutions ◽  
Flow Components

Potential flow equations are converted to ordinary differential equations by the Galerkin approach in which velocity and pressure potential functions are expanded in terms of closed-form solutions to Laplace’s Equation. The reduced number of generalized coordinates in a Galerkin approach gives advantages in real-time simulations, preliminary design, and dynamic eigenvalue analysis for aeroelasticity. Net mass injection from rotor sources is expected to occur in some situations, but cannot be treated by previous models. It is included in the present formulation. In this paper, frequency response due to pressure distributions corresponding to net mass flow in both axial and skew-angle flight are given. These results are compared with exact solutions obtained by the approach of a convolution integral. A brief analysis is also included with respect to numerical simulations of the Associated Legendre Functions, in which it demonstrates that net mass flow components are extremely sensitive to the recursive process of seeking Associated Legendre Functions.

Analytical Modeling for the Shape Control of an Adaptive Composite Satellite Dish

2008 ◽  
Author(s):  
Su Yan ◽  
Mehrdad N. Ghasemi-Nejhad
Keyword(s):  
Shape Control ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Legendre Functions ◽  
Spherical Coordinate ◽  
Associated Legendre Functions ◽  
Satellite Dish ◽  
Active Composite ◽  
Adaptive Composite

A three-dimensional dynamic analysis of a satellite dish in spherical coordinate system is investigated. The method of separation of variables is employed to obtain the explicit 3D solution of the partial differential governing equation of the active composite satellite dish (ACSD). Then, the mode shape functions are expanded as a combination of periodic functions, associated Legendre functions, and spherical Bessel functions. The validation of the theoretical model is performed by comparing the developed analytical mode shapes with finite element method (FEM) mode shapes. Also, using the developed analytical model, the disturbance observer (DOB) controller is employed for the ACSD shape control. The numerical results show that, by employing the DOB controller, more accurate shape control of the satellite dish is achieved and less control energy is consumed, when piezoelectric actuator patches are placed on their optimal locations.

State-space inflow modelling for lifting rotors with mass injection

2004 ◽  
Vol 108 (1085) ◽  
pp. 333-344 ◽  
Author(s):  
K. Yu ◽  
D. A. Peters
Keyword(s):  
State Space ◽  
Potential Flow ◽  
State Space Model ◽  
Legendre Functions ◽  
Skew Angle ◽  
Mass Source ◽  
Associated Legendre Functions ◽  
Space Model ◽  
Mass Injection ◽  
Mass Influx

Abstract In the field of rotorcraft dynamics, it is significant that the induced inflow field is well understood and modeled. A large number of methodologies have been developed in the past years, among which the state-space model is recognised for its advantage in real-time simulation, preliminary design, and dynamic eigenvalue analysis. Recent studies have shown success in representing the induced flow field everywhere above the rotor plane even with mass source terms on the disk as long as they have zero net flux of mass injection when integrated over the disk. Nevertheless, non-zero net mass influx is expected in numerous situations, such as ground effect, tip drive rotors, etc; and the incapability of previous models limits the utilisation of the methodology in these cases. This work presents an extended potential-flow, state-space model derived from the potential-flow momentum equation by means of a Galerkin approach. The induced velocity and pressure perturbation are expanded in terms of closed-form, time-dependent coefficients and space-dependent associated Legendre functions and harmonics. Non-zero net mass flux terms are represented by the involvement of associated Legendre functions with equal degrees and orders. Validation, as well as discrepancies, of the inclusion of such terms is investigated. Numerical simulation of frequency response in axial and skew-angle flight is presented and compared with exact solutions obtained by the convolution integral. Also the study shows that, unlike other pressure distribution responses, non-zero mass influx exhibits a high sensitivity to the choice of the number of states in the velocity expansion. Error analyses are performed to show this sensitivity.

THREE-DIMENSIONAL INFINITE ELEMENTS BASED ON A TREFFTZ FORMULATION

2001 ◽  
Vol 09 (02) ◽  
pp. 381-394 ◽  
Author(s):  
ISAAC HARARI ◽  
PARAMA BARAI ◽  
PAUL E. BARBONE ◽  
MICHAEL SLAVUTIN
Keyword(s):  
Separation Of Variables ◽  
Outer Boundary ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Singular Behavior ◽  
Associated Legendre Functions ◽  
Exterior Problems ◽  
Infinite Elements ◽  
Time Harmonic

Three-dimensional infinite elements for exterior problems of time-harmonic acoustics are developed. The infinite elements mesh only the outer boundary of the finite element domain and need not match the finite elements on the interface. A four-noded infinite element, based on separation of variables in spherical coordinates, is presented. Singular behavior of associated Legendre functions at the poles is circumvented. Numerical results validate the good performance of this approach.

Thermal Characterization of Interlayer Microfluidic Cooling of Three-Dimensional Integrated Circuits With Nonuniform Heat Flux

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Yoon Jo Kim ◽  
Yogendra K. Joshi ◽  
Andrei G. Fedorov ◽  
Young-Joon Lee ◽  
Sung-Kyu Lim
Keyword(s):  
Integrated Circuits ◽  
Mass Flow Rate ◽  
Thermal Management ◽  
Mass Flow ◽  
Flow Rate ◽  
Single Phase ◽  
Hot Spot ◽  
Three Dimensional ◽  
Two Phase ◽  
Two Phase Cooling

It is now widely recognized that the three-dimensional (3D) system integration is a key enabling technology to achieve the performance needs of future microprocessor integrated circuits (ICs). To provide modular thermal management in 3D-stacked ICs, the interlayer microfluidic cooling scheme is adopted and analyzed in this study focusing on a single cooling layer performance. The effects of cooling mode (single-phase versus phase-change) and stack/layer geometry on thermal management performance are quantitatively analyzed, and implications on the through-silicon-via scaling and electrical interconnect congestion are discussed. Also, the thermal and hydraulic performance of several two-phase refrigerants is discussed in comparison with single-phase cooling. The results show that the large internal pressure and the pumping pressure drop are significant limiting factors, along with significant mass flow rate maldistribution due to the presence of hot-spots. Nevertheless, two-phase cooling using R123 and R245ca refrigerants yields superior performance to single-phase cooling for the hot-spot fluxes approaching ∼300 W/cm2. In general, a hybrid cooling scheme with a dedicated approach to the hot-spot thermal management should greatly improve the two-phase cooling system performance and reliability by enabling a cooling-load-matched thermal design and by suppressing the mass flow rate maldistribution within the cooling layer.

A Three-Dimensional Viscous Transonic Inverse Design Method

2000 ◽  
Author(s):  
W. T. Tiow ◽  
M. Zangeneh
Keyword(s):  
Design Method ◽  
Flow Analysis ◽  
Three Dimensional ◽  
Viscous Effects ◽  
Flow Equations ◽  
Flow Solution ◽  
Turbomachinery Blades ◽  
A Cell ◽  
Inverse Design Method ◽  
Euler Solver

The development and application of a three-dimensional inverse methodology is presented for the design of turbomachinery blades. The method is based on the mass-averaged swirl, rV~θ distribution and computes the necessary blade changes directly from the discrepancies between the target and initial distributions. The flow solution and blade modification converge simultaneously giving the final blade geometry and the corresponding steady state flow solution. The flow analysis is performed using a cell-vertex finite volume time-marching algorithm employing the multistage Runge-Kutta integrator in conjunction with accelerating techniques (local time stepping and grid sequencing). To account for viscous effects, dissipative forces are included in the Euler solver using the log-law and mixing length models. The design method can be used with any existing solver solving the same flow equations without any modifications to the blade surface wall boundary condition. Validation of the method has been carried out using a transonic annular turbine nozzle and NASA rotor 67. Finally, the method is demonstrated on the re-design of the blades.

A Three-Dimensional Finite Difference Solution for the Thermal Stresses in Railcar Wheels

1969 ◽  
Vol 91 (3) ◽  
pp. 891-896 ◽  
Author(s):  
G. E. Novak ◽  
B. J. Eck
Keyword(s):  
Computer Program ◽  
Thermal Stresses ◽  
Three Dimensional ◽  
Transient Temperature ◽  
Shear Stresses ◽  
Radial Temperature ◽  
Closed Form Solutions ◽  
Brake Application ◽  
History Of ◽  
Thin Disks

A numerical solution is presented for both the transient temperature and three-dimensional stress distribution in a railcar wheel resulting from a simulated emergency brake application. A computer program has been written for generating thermoelastic solutions applicable to wheels of arbitrary contour with temperature variations in both axial and radial directions. The results include the effect of shear stresses caused by the axial-radial temperature gradients and the high degree of boundary irregularity associated with this type of problem. The program has been validated by computing thermoelastic solutions for thin disks and long cylinders; the computed values being in good agreement with the closed form solutions. Currently, the computer program is being extended to general stress solutions corresponding to the transient temperature distributions obtained by simulated drag brake applications. When this work is completed, it will be possible to synthesize the thermal history of a railcar wheel and investigate the effects of wheel geometry in relation to thermal fatigue.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

scholarly journals Through-Flow Analysis for Axial-Stage Design Including Streamline-Slope Effects

1993 ◽  
Author(s):  
Theodosios Korakianitis ◽  
Dequan Zou
Keyword(s):  
Mass Flow ◽  
Three Dimensional ◽  
Momentum Equation ◽  
Total Enthalpy ◽  
Streamline Curvature ◽  
Flow Balance ◽  
Through Flow ◽  
Blade Row ◽  
Iteration Loop ◽  
Predictor Corrector

This paper presents a new method to design (or analyze) subsonic or supersonic axial compressor and turbine stages and their three-dimensional velocity diagrams from hub to tip by solving the three-dimensional radial-momentum equation. Some previous methods (matrix through-flow based on the streamfunction approach) can not handle locally supersonic flows, and they are computationally intensive when they require the inversion of large matrices. Other previous methods (streamline curvature) require two nested iteration loops to provide a converged solution: an outside iteration loop for the mass-flow balance; and an inside iteration loop to solve the radial momentum equation at each flow station. The present method is of the streamline-curvature category. It still requires the iteration loop for the mass-flow balance, but the radial momentum equation at each flow station is solved using a one-pass numerical predictor-corrector technique, thus reducing the computational effort substantially. The method takes into account the axial slope of the streamlines. Main design characteristics such as the mass-flow rate, total properties at component inlet, hub-to-tip ratio at component inlet, total enthalpy change for each stage, and the expected efficiency of each streamline at each stage are inputs to the method. Other inputs are the radial position and axial velocity component at one surface of revolution through the axial stages. These can be provided for either the hub, or the mean, or the tip location of the blading. In addition the user specifies the azimuthal deflection of the flow from the axial direction at each radius (or as a function of radius) at each blade row inlet and outlet. By construction the method eliminates radial variations of total enthalpy (work) and entropy at each blade row inlet and outlet. In an alternative formulation enthalpy variations across radial positions at each axial station are included in the analysis. The remaining three-dimensional velocity diagrams from hub to tip, and the radial location of the remaining streamlines, are obtained by solving the momentum equation using a predictor-corrector method. Examples for one turbine and one compressor design are included.

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