scholarly journals Inverse Method for Turbomachine Blades Using Existing Time-Marching Techniques

1994 ◽  
Author(s):  
T. Dang ◽  
V. Isgro
Keyword(s):  
Euler Equations ◽  
Inverse Method ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Tangential Velocity ◽  
Steady State Solution ◽  
Slip Boundary ◽  
Time Stepping ◽  
Time Marching ◽  
Blade Geometry

A newly-developed inverse method for the design of turbomachine blades using existing time-marching techniques for the numerical solutions of the unsteady Euler equations is proposed. In this inverse method, the pitch-averaged tangential velocity (or the blade loading) is the specified quantity, and the corresponding blade geometry is Iteratively sought after. The presence of the blades are represented by a periodic array of discrete body forces which are included in the equations of motion. A four-stage Runge-Kutta time-stepping scheme is used to march a finite-volume formulation, of the unsteady Euler equations to a steady-state solution. Modification of the blade geometry during this time marching process is achieved using the slip boundary conditions on the blade surfaces. This method is demonstrated for the design of infinitely-thin cascaded blades in the subsonic, transonic, and supersonic flow regimes. Results are validated using an Euler analysis method and are compared against those obtained using a similar inverse method. Excellent agreement in the results are obtained between these different approaches.

Throughflow Method for Turbomachines Applicable for All Flow Regimes

1997 ◽  
Vol 119 (2) ◽  
pp. 256-262 ◽  
Author(s):  
S. V. Damle ◽  
T. Q. Dang ◽  
D. R. Reddy
Keyword(s):  
Body Force ◽  
Equations Of Motion ◽  
Tangential Velocity ◽  
Matrix Methods ◽  
Streamline Curvature ◽  
Time Stepping ◽  
Viscous Losses ◽  
Blade Shape ◽  
Flow Quantity ◽  
Analysis Mode

A new axisymmetric throughflow method for analyzing and designing turbomachines is proposed. This method utilizes body-force terms to represent blade forces and viscous losses. The resulting equations of motion, which include these body-force terms, are cast in terms of conservative variables and are solved using a finite-volume time-stepping scheme. In the inverse mode, the swirl schedule in the bladed regions (i.e., the radius times the tangential velocity rVθ) is the primary specified flow quantity, and the corresponding blade shape is sought after. In the analysis mode, the blade geometry is specified and the flow solution is computed. The advantages of this throughflow method compared to the current family of streamline curvature and matrix methods are that the same code can be used for subsonic/transonic/supersonic throughflow velocities, and the proposed method has a shock capturing capability. This method is demonstrated for designing a supersonic throughflow fan stage and a transonic throughflow turbine stage.

Throughflow Method for Turbomachines Applicable for All Flow Regimes

1995 ◽  
Author(s):  
S. V. Damle ◽  
T. Q. Dang ◽  
D. R. Reddy
Keyword(s):  
Body Force ◽  
Equations Of Motion ◽  
Tangential Velocity ◽  
Matrix Methods ◽  
Streamline Curvature ◽  
Time Stepping ◽  
Viscous Losses ◽  
Blade Shape ◽  
Flow Quantity ◽  
Analysis Mode

A new axisymmetric throughflow method for analyzing and designing turbomachines is proposed. This method utilizes body-force terms to represent blade forces and viscous losses. The resulting equations of motion, which include these body-force terms, are casted in terms of conservative variables and are solved using a finite-volume time-stepping scheme. In the inverse mode, the swirl schedule in the bladed regions (i.e. the radius times the tangential velocity rVθ) is the primary specified flow quantity, and the corresponding blade shape is sought after. In the analysis mode, the blade geometry is specified and the flow solution is computed. The advantages of this throughflow method compared to the current family of streamline curvature and matrix methods are that the same code can be used for subsonic/transonic/supersonic throughflow velocities, and the proposed method has a shock capturing capability. This method is demonstrated for designing a supersonic throughflow fan stage and a transonic throughflow turbine stage.

A 3D Approximate Riemann Solver for the Euler Equations Using Flux Splitting Schemes With a Robust Multigrid

2016 ◽  
Author(s):  
Hong-Sik Im
Keyword(s):  
Euler Equations ◽  
Riemann Solver ◽  
Riemann Solvers ◽  
Approximate Riemann Solver ◽  
Time Stepping ◽  
Flux Vector Splitting ◽  
Time Marching ◽  
Convergence To Steady State ◽  
Flux Difference ◽  
Robust Multigrid

An explicit 3D approximate Riemann solver for the Euler equations is proposed using the famous shock capturing schemes with a simple cell vertex based multigrid method. A multistage Runge-Kutta time marching scheme with a local time stepping is used to achieve fast convergence to steady state. A Roe’s flux difference splitting, AUSM+, Van Leer and Steger-Warming’s flux vector splitting are implemented as base Riemann solvers with a third order flux reconstruction. It is shown that the proposed Riemann solvers accurately capture the shocks as well as reduce CPU time significantly with new multigrid.

scholarly journals Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

2021 ◽  
Vol 11 (8) ◽  
pp. 3421
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang
Keyword(s):  
Porous Media ◽  
Time Domain ◽  
Numerical Solutions ◽  
Space Time ◽  
Two Dimensional ◽  
Transient Flows ◽  
Layered Porous Media ◽  
Time Marching ◽  
Time Basis ◽  
Marching Scheme

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

The Dynamic Behaviour of Passively Compensated, Hydrostatic Journal Bearings with Various Numbers of Recesses

1976 ◽  
Vol 18 (6) ◽  
pp. 292-302 ◽  
Author(s):  
P. B. Davies
Keyword(s):  
Dynamic Behaviour ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
Flow Equation ◽  
External Excitation ◽  
Signal Flow Graphs ◽  
Flow Compensation ◽  
Circular Functions ◽  
Flow Graphs

A previously established small-perturbation analysis is developed to express the unsteady-state continuity-of-flow equation for an isolated recess in a passively compensated, multirecess, hydrostatic journal bearing in terms of generalized co-ordinates. The concise form of this equation enables motion of the shaft about the concentric position to be described by equations which are derived in closed form for bearings with orifice, capillary or constant flow compensation and any number of recesses. These equations of motion, and hence the expressions for the receptances which describe the response of a bearing to external excitation, are shown to be of exactly the same form for all bearings of the type considered. Furthermore, the damping ratio and natural frequency in any particular case are determined by a single dynamic constant which is shown to be equal to a linear combination of circular functions and a limited number of coefficients which may be found explicitly by routine use of signal flow graphs. The results of the analysis, which is exact within the stated assumptions, are compared with those of other workers and the steady-state solution of the equations of motion is shown to give an expression for static stiffness which is useful for design purposes. Numerical values of the dynamic constant for bearings with between 3 and 20 recesses are given graphically.

scholarly journals Use of an Inverse Method for the Design of High Efficiency Compressor and Turbine Blades With Large Change in Radius

1984 ◽  
Author(s):  
G. Meauzé ◽  
A. Lesain
Keyword(s):  
Inverse Method ◽  
High Efficiency ◽  
Radial Flow ◽  
Turbine Blades ◽  
Large Change ◽  
Airfoil Optimization ◽  
Time Marching ◽  
Flow Compressor ◽  
Compressor And Turbine Blades ◽  
Made In

Extension of the time-marching computations of flows in 2-D blade cascades to the case of cascades with variable radius and stream tube thickness. One of the specific cases analyzed is that of purely radial cascades. Direct and inverse calculations are made, in non-viscous subsonic or supersonic flows, with or without shock waves. Examples of the design of high efficiency airfoil optimization for radial flow compressor rotors or Stators or inward flow turbine inlet guide vanes are presented.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

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