Multibladerow Forced Response Modeling in Axial-Flow Core Compressors

2005 ◽  
Vol 129 (2) ◽  
pp. 412-420 ◽  
Author(s):  
M. Vahdati ◽  
A. I. Sayma ◽  
M. Imregun ◽  
G. Simpson
Keyword(s):  
Structural Model ◽  
Stokes Equations ◽  
Axial Flow ◽  
Forced Response ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Engine Order ◽  
Grid Points ◽  
Response Modeling

This paper describes the formulation and application of an advanced numerical model for the simulation of blade-passing and low-engine order forced response in turbomachinery core compressors. The Reynolds averaged Navier–Stokes equations are used to represent the flow in a nonlinear time-accurate fashion on unstructured meshes of mixed elements. The structural model is based on a standard finite-element representation. The fluid mesh is moved at each time step according to the structural motion so that changes in blade aerodynamic damping and flow unsteadiness can be accommodated automatically. A whole-annulus 5-bladerow forced response calculation, where three upstream and one downstream bladerows were considered in addition to the rotor bladerow of interest, was undertaken using over 20 million grid points. The results showed not only some potential shortcomings of equivalent 2-bladerow computations for the determination of the main blade-passing forced response, but also revealed the potential importance of low engine-order harmonics. Such harmonics, due to stator blade number differences, or arising from common symmetric sectors, can only be taken into account by including all stator bladerows of interest. The low engine-order excitation that could arise from a blocked passage was investigated next. It was shown that high vibration response could arise in such cases.

Turbine forced response prediction using an integrated non-linear analysis

2000 ◽  
Vol 214 (1) ◽  
pp. 45-60 ◽  
Author(s):  
A I Sayma ◽  
M Vahdati ◽  
M Imregun
Keyword(s):  
Stokes Equations ◽  
Prediction Method ◽  
Response Prediction ◽  
Forced Response ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Engine Order ◽  
Lp Turbine ◽  
Nozzle Guide

The forced response due to flow defects caused by the upstream blade rows is predicted for two turbines: intermediate pressure (IP) and low pressure (LP). The prediction method is based on an advanced numerical tool where the compressible viscous flow field is modelled by solving Favre-averaged Navier-Stokes equations with the Baldwin and Barth turbulence model. The flow solution is coupled to a modal model of the structure and information is exchanged every time step between the fluid and the structural domains. The hybrid unstructured mesh is moved at each time step to follow the structural motion using a spring analogy. For the IP turbine, the method was used to rank two different designs of nozzle guide vanes. For the LP turbine, special emphasis was placed on predicting vibration amplitudes due to high and low engine order excitations. Predictions and measurements were found to be in good agreement for both turbines. Due to insufficient experimental data, it was difficult to assess the accuracy of the low engine order computations, although it was shown that the model was capable of undertaking such a task.

A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle

2017 ◽  
Vol 32 (4) ◽  
Author(s):  
Alexander Danilov ◽  
Alexander Lozovskiy ◽  
Maxim Olshanskii ◽  
Yuri Vassilevski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Left Ventricle ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Computed Tomography Images ◽  
Time Dependent Domain ◽  
Element Method

AbstractThe paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Forced Response Assessment Using Modal Force Based Indicator Functions

2008 ◽  
Author(s):  
M. Vahdati ◽  
C. Breard ◽  
G. Simpson ◽  
M. Imregun
Keyword(s):  
Finite Element ◽  
Structural Model ◽  
Stokes Equations ◽  
Linear Time ◽  
Response Assessment ◽  
Element Model ◽  
Forced Response ◽  
Navier Stokes ◽  
Modal Force ◽  
Indicator Functions

This paper will focus on core-compressor forced response with the aim to develop two design criteria, the so-called chordwise cumulative modal force and heightwise cumulative force, to assess the potential severity of the vibration levels from the correlation between the unsteady pressure distribution on the blade’s surface and the structural modeshape. It is also possible to rank various blade designs since the proposed criterion is sensitive to changes in both unsteady aerodynamic loads and the vibration modeshapes. The proposed methodology was applied to a typical core-compressor forced response case for which measured data were available. The Reynolds-averaged Navier-Stokes equations were used to represent the flow in a non-linear time-accurate fashion on unstructured meshes of mixed elements. The structural model was based on a standard finite element representation from which the vibration modes were extracted. The blade flexibility was included in the model by coupling the finite element model to the unsteady flow model in a time-accurate fashion. A series of numerical experiments were conducted by altering the stator wake and using the proposed indicator functions to minimize the rotor response levels. It was shown that a fourfold response reduction was possible for a certain mode with only a minor modification of the blade.

Annular Axial Flow-Induced Vibration of Cylindrical Shells by Flu¨gge’s Shell Theory

2006 ◽  
Author(s):  
Katsuhisa Fujita ◽  
Makoto Katou
Keyword(s):  
Shell Theory ◽  
Stokes Equations ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Navier Stokes ◽  
Complex Eigenvalue ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Flowing Fluid ◽  
Narrow Passage

The unstable phenomena of thin cylindrical shells subjected to annular axial flow are investigated. In this paper, the analytical model is composed of an elastic axisymmetric shell and a rigid one which are arranged co-axially. Considering the fluid structure interaction between shells and fluid flowing through an annular narrow passage, the coupled equation of motion is derived using Flu¨gge’s shell theory and Navier-Stokes equations. The unstable phenomena of thin cylindrical shells are clarified by using the root locus based on the complex eigenvalue analysis. The numerical parameter studies on the shells with a freely supported end and a rigid one, and with both simply supported ends, are performed taking the dimensins of shells, the characteristics of flowing fluid so on as parameters. The influence of these parameters on the threshold of instability of the coupled vibration between thin cylindrical shells and annular axial flowing fluid are investigated and discussed.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

Rotordynamic characterization of tilting-pad bearings with eight pads in vertical rotors

2021 ◽  
pp. 1-20
Author(s):  
David J Rondon ◽  
Gudeta Berhanu Benti ◽  
Jan-Olov Aidanpää ◽  
Rolf Gustavsson
Keyword(s):  
Stokes Equations ◽  
Forced Response ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Mean Values ◽  
Stationary Coordinate System ◽  
Tilting Pad ◽  
Tilting Pad Bearings ◽  
Stiffness And Damping

Abstract It has been documented that stiffness and damping for a four-pad bearing are dependent not only the magnitude of the load but also on the position of the rotor in the bearing. However, 8-pad bearings are not commonly employed on horizontal turbines, and the presence of several pads in the bearing will decisively affect the dynamics of the system. This paper evaluates the stiffness and damping coefficients of tilting-pad bearings with eight pads and explore the main frequencies acting on the forced response of a vertical rotor. The bearing properties were modeled as a function of eccentricity and position in the stationary coordinate system by Navier-Stokes equations whose results are taken from commercial software. The simulated unbalanced response is compared to experimental results; the changing position of the shaft produces a periodic stiffness and damping, which is dependent on the number of pads. Cross-coupled coefficients influence is discussed, showing that their absence makes an accurate model for the mean values. The results indicate that simulation of vertical rotors with 8-pad bearings can be simplified which allow more effective simulations and dynamic analysis.

scholarly journals Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections

2020 ◽  
Vol 10 (22) ◽  
pp. 7952
Author(s):  
Qiang Wang ◽  
Boran Zhang ◽  
Pengyao Yu ◽  
Guangzhao Li ◽  
Zhijiang Yuan
Keyword(s):  
Stokes Equations ◽  
Water Entry ◽  
Navier Stokes ◽  
Hydrodynamic Characteristics ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Surface Evolution ◽  
Mesh Method ◽  
The Difference ◽  
Dynamics Method

The bow-flared section may be simplified in the prediction of slamming loads and whipping responses of ships. However, the difference of hydrodynamic characteristics between the water entry of the simplified sections and that of the original section has not been well documented. In this study, the water entry of several different bow-flared sections was numerically investigated using the computational fluid dynamics method based on Reynolds-averaged Navier–Stokes equations. The motion of the grid around the section was realized using the overset mesh method. Reasonable grid size and time step were determined through convergence studies. The application of the numerical method in the water entry of bow-flared sections was validated by comparing the present predictions with previous numerical and experimental results. Through a comparative study on the water entry of one original section and three simplified sections, the influences of simplification of the bow-flared section on hydrodynamic characteristics, free surface evolution, pressure field, and impact force were investigated and are discussed here.

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

1969 ◽  
Vol 37 (4) ◽  
pp. 727-750 ◽  
Author(s):  
Gareth P. Williams
Keyword(s):  
Poisson Equation ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Wave Flow ◽  
Trigonometric Interpolation ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Poisson Equation

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

scholarly journals Fast solvers for finite difference approximations for the stokes and navier-stokes equations

1996 ◽  
Vol 38 (2) ◽  
pp. 274-290 ◽  
Author(s):  
Dongho Shin ◽  
John C. Strikwerda
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Linear Equations ◽  
Computational Effort ◽  
Navier Stokes ◽  
Fast Solvers ◽  
Navier Stokes Equations ◽  
Finite Difference Approximations ◽  
Grid Points ◽  
First Time

AbstractWe consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

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