Multistage turbomachinery flow solutions using three-dimensional implicit Euler method

1993 ◽  
Author(s):  
SHIH CHEN ◽  
GEORGE PRUEGER
Keyword(s):  
Three Dimensional ◽  
Euler Method ◽  
Implicit Euler Method

scholarly journals Implicit Euler Method for Three-Dimensional Turbomachinery Flow Calculation

1993 ◽  
Author(s):  
Shih H. Chen ◽  
Anthony H. Eastland
Keyword(s):  
Three Dimensional ◽  
Prediction Method ◽  
Minimum Cost ◽  
Inviscid Flow ◽  
Turnaround Time ◽  
Euler Method ◽  
Second Order ◽  
Approximate Factorization ◽  
Implicit Euler Method ◽  
Numerical Instabilities

A compressible three-dimensional implicit Euler solution method for turbomachinery flows has been developed. The goal of the present study is to develop an efficient and reliable method that can be used to replace the semi-empirical, semi-analytical quasi-three-dimensional turbomachinery flow prediction method currently being used for multi-stage turbomachinery design at early design stages. Currently, a methodology has been developed based on an inviscid flow model (Euler solver) and tested on single blade rows for validation. The method presented here is derived from the Beam and Warming implicit approximate factorization (AF) finite difference algorithm. To avoid high frequency numerical instabilities associated with the use of central differencing schemes to obtain a spatial second order accuracy, a combined explicit and implicit artificial dissipation model is adopted. This model consists of a second order implicit dissipation and mixed second/fourth order explicit dissipation terms. A Cartesian coordinate H-grid generated by a three-dimensional interactive grid generator developed by Beach is used. Results for SSME High Pressure Fuel Turbine are presented and the comparison with experimental data is discussed. The use of the present implicit Euler method and the three-dimensional turbomachinery interactive grid generator shows that turnaround time could be as short as one day using a workstation. This allows the designers to explore optimal design configurations at minimum cost.

Nonlinear dynamic analysis of worn oil-lubricated rolling bearings

2020 ◽  
Vol 234 (2) ◽  
pp. 214-221
Author(s):  
Xue-Qian Fang ◽  
Fu-Ning Liu ◽  
Shao-Pu Yang
Keyword(s):  
Film Thickness ◽  
Nonlinear Dynamic ◽  
Significant Variation ◽  
Nonlinear Dynamic Analysis ◽  
Euler Method ◽  
Rolling Speed ◽  
Dynamic Equations ◽  
Rolling Bearings ◽  
Numerical Examples ◽  
Implicit Euler Method

Based on the elastohydrodynamic theory, the nonlinear dynamic behavior of worn oil-lubricated rolling bearings is explored, and the dynamic response including the effect of trajectory of the axis center, the accelerated speed, and the film thickness is analyzed. The worn model is represented by the worn depth. The discrete iterative method and implicit Euler method are combined to solve the dynamic equations. In numerical examples, the trajectory of the axis center, the accelerated speed and the film thickness under different worn depths are discussed. It is found that the stabilized point shows significant variation with the worn depth, and the wear effect is also quite related with the rolling speed. The trajectory of the axis center of worn bearing subjected to a step load is also examined in detail.

scholarly journals On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

2015 ◽  
Vol 52 (02) ◽  
pp. 323-338 ◽  
Author(s):  
Mihály Kovács ◽  
Stig Larsson ◽  
Fredrik Lindgren
Keyword(s):  
Gaussian Noise ◽  
Smooth Boundary ◽  
Spatial Domain ◽  
Euler Method ◽  
Euler Approximation ◽  
Implicit Euler Method ◽  
Cahn Equation ◽  
Backward Euler

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt γ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.

scholarly journals Heat Transfer during Steaming of Bread

2014 ◽  
Vol 10 (4) ◽  
pp. 613-623 ◽  
Author(s):  
Victoria K. Ananingsih ◽  
Edda Y. L. Sim ◽  
Xiao Dong Chen ◽  
Weibiao Zhou
Keyword(s):  
Heat Transfer ◽  
Three Dimensional ◽  
Euler Method ◽  
Heating Process ◽  
Temperature Profiles ◽  
Transfer System ◽  
Physicochemical Changes ◽  
Steamed Bread ◽  
Predicted Values ◽  
Heat Transfer System

Abstract Understanding of heat transfer during steaming is important to optimize the processing of steamed bread and to produce desired qualities in the final product. Physicochemical changes occur during steaming of the dough which might be impacted upon by the heat transfer system. In this study, a mathematical model was developed to describe the heat transfer system in the bread being steaming throughout the heating process. The Forward Euler method was employed for solving the three-dimensional partial differential equation set for heat transfer to produce temperature profiles at a number of individual locations in the steamed bread during its steaming. All the comparisons between the model-predicted values and the experimental results produced root mean square error values ranged from 1.391 to 3.545 and R2 values of all greater than 0.93. Therefore, it is confirmed that the model has a good performance and can be used to predict temperature profiles in the bread during steaming.

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