Analysis for Incompressible Flow in Annular Pressure Seals

1992 ◽  
Vol 114 (3) ◽  
pp. 431-438 ◽  
Author(s):  
F. Simon ◽  
J. Freˆne
Keyword(s):  
Incompressible Flow ◽  
Stokes Equations ◽  
Steady State Solution ◽  
Navier Stokes ◽  
Zeroth Order ◽  
Navier Stokes Equations ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
The Moment ◽  
Flow Variables

An analysis is developed to calculate the static and dynamic characteristics of annular eccentric seals. Effects of inertia forces in the film, tapered geometry and rotor misalignment are taken into account. Derivation of the governing equations for incompressible flow is based on the Navier-Stokes equations, the continuity equation and a turbulence model using the nonlinear analysis developed by Elrod and Ng. The inlet boundary conditions define the initial swirl and the pressure drop due to the fluid acceleration. Perturbation of the flow variables yields a set of zeroth-order and first-order equations. Integration of the zeroth-order equations yields the steady-state solution which defines the seal leakage, the static load and the moment of misalignment. The eccentric and misaligned rotordynamic coefficients are obtained by integration of the first-order pressure equations. Comparisons are made between the stiffness, damping and inertia coefficients derived herein and both theoretical results based on other models and experimental data which were previously published.

Calculation of Rotordynamic Coefficients and Leakage for Annular Gas Seals by Means of Finite Difference Techniques

1989 ◽  
Vol 111 (3) ◽  
pp. 545-552 ◽  
Author(s):  
R. Nordmann ◽  
F. J. Dietzen ◽  
H. P. Weiser
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Zeroth Order ◽  
Navier Stokes Equations ◽  
First Order ◽  
Fluid Forces ◽  
Rotordynamic Coefficients ◽  
Gas Seals ◽  
Finite Difference Techniques

The compressible flow in a seal can be described by the Navier-Stokes equations in connection with a turbulence model (k–ε model) and an energy equation. By introducing a perturbation analysis in these differential equations we obtain zeroth order equations for the centered position and first order equations for small motions of the shaft about the centered position. These equations are solved by a finite difference technique. The zeroth order equations describe the leakage flow. Integrating the pressure solution of the first order equations yields the fluid forces and the rotordynamic coefficients, respectively.

Rotordynamic Coefficients of Circumferentially-Grooved Liquid Seals Using the Averaged Navier-Stokes Equations

1997 ◽  
Vol 119 (3) ◽  
pp. 556-567 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Freˆne
Keyword(s):  
Coordinate Transformation ◽  
Stokes Equations ◽  
Turbulent Viscosity ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
First Order ◽  
Rotordynamic Coefficients ◽  
High Reynolds ◽  
Good Agreement

The paper presents a method to calculate the rotordynamic coefficients of circumferentially-grooved liquid seals operating in centered position and turbulent flow regimes. The method is based on the integration of the averaged Navier-Stokes equations and uses a coordinate transformation proposed by Dietzen and Nordmann (1987). The effect of the coordinate transformation on the components of the stress tensor is included in the first order transport equations. To ensure grid independent solutions, numerical boundary conditions for the first-order velocities were formulated using the logarithmic law. The perturbation of the turbulent viscosity was also considered. A pressure recovery effect at the exit section was included in the first order mathematical model. The method is validated by calculations for straight and circumferentially-grooved seals. Comparisons with experimental and theoretical results show a good agreement for straight seals and for seals with few grooves, and a reasonable agreement for severe industrial cases (high Reynolds numbers and large number of grooves).

Rotordynamic Coefficients of Labseals for Turbines: Comparing CFD Results With Experimental Data on a Comb-Grooved Labyrinth

2005 ◽  
Author(s):  
Joachim Schettel ◽  
Martin Deckner ◽  
Klaus Kwanka ◽  
Bernd Lu¨neburg ◽  
Rainer Nordmann
Keyword(s):  
Stokes Equations ◽  
Labyrinth Seal ◽  
Navier Stokes ◽  
Test Rig ◽  
Navier Stokes Equations ◽  
Detailed Model ◽  
Identification Methods ◽  
Rotating Speed ◽  
Flow Models ◽  
Rotordynamic Coefficients

The main goal of this paper is to improve identification methods for rotordynamic coefficients of labseals for turbines. This aim was achieved in joint effort of the Technische Universita¨t Mu¨nchen, working on experimental identification methods for rotordynamic coefficients, the University of Technology, Darmstadt, working on prediction methods, and Siemens AG, realizing the results. The paper focuses on a short comb-grooved labyrinth seal. Short labseals, amongst others the above mentioned comb-grooved labyrinth, were examined. by means of a very accurately measuring test rig. The rotor was brought into statically eccentric positions relative to the stator, in order to measure the circumferential pressure distribution as a function of pressure, rotating speed and entrance swirl. The data collected were used to validate results obtained with a numerical method. The theoretical approach is based on a commercial CFD tool, which solves the Navier Stokes equations using numerical methods. As a result, a detailed model of the flow within the test rig is produced. The efforts of computation here are greater than when compared with the likewise wide-spread Bulk flow models, however improved accuracy and flexibility is expected. As the validation of the model is successful, it could then be used to gain further insight in the flow within the seal, and to understand the results better. This showed that rotordynamic coefficients of labseals gained from different test rigs are not necessarily comparable.

scholarly journals Mathematical Model for the Electrokinetic Instability of Electrolyte Flow

2019 ◽  
Vol 224 ◽  
pp. 02003
Author(s):  
Andrey Shobukhov
Keyword(s):  
Electric Potential ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Steady State Solution ◽  
Stability Loss ◽  
Navier Stokes ◽  
Ion Distribution ◽  
Navier Stokes Equations ◽  
Dilute Aqueous Solution

We study a one-dimensional model of the dilute aqueous solution of KCl in the electric field. Our model is based on a set of Nernst-Planck-Poisson equations and includes the incompressible fluid velocity as a parameter. We demonstrate instability of the linear electric potential variation for the uniform ion distribution and compare analytical results with numerical solutions. The developed model successfully describes the stability loss of the steady state solution and demonstrates the emerging of spatially non-uniform distribution of the electric potential. However, this model should be generalized by accounting for the convective movement via the addition of the Navier-Stokes equations in order to substantially extend its application field.

scholarly journals A multilayer shallow model for dry granular flows with the -rheology: application to granular collapse on erodible beds

2016 ◽  
Vol 798 ◽  
pp. 643-681 ◽  
Author(s):  
E. D. Fernández-Nieto ◽  
J. Garres-Díaz ◽  
A. Mangeney ◽  
G. Narbona-Reina
Keyword(s):  
Transient Flow ◽  
Stokes Equations ◽  
Low Cost ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Runout Distance ◽  
Multilayer Model ◽  
First Order ◽  
Constant Friction ◽  
Erodible Beds

In this work we present a multilayer shallow model to approximate the Navier–Stokes equations with the ${\it\mu}(I)$-rheology through an asymptotic analysis. The main advantages of this approximation are (i) the low cost associated with the numerical treatment of the free surface of the modelled flows, (ii) the exact conservation of mass and (iii) the ability to compute two-dimensional profiles of the velocities in the directions along and normal to the slope. The derivation of the model follows Fernández-Nieto et al. (J. Comput. Phys., vol. 60, 2014, pp. 408–437) and introduces a dimensional analysis based on the shallow flow hypothesis. The proposed first-order multilayer model fully satisfies a dissipative energy equation. A comparison with steady uniform Bagnold flow – with and without the sidewall friction effect – and laboratory experiments with a non-constant normal profile of the downslope velocity demonstrates the accuracy of the numerical model. Finally, by comparing the numerical results with experimental data on granular collapses, we show that the proposed multilayer model with the ${\it\mu}(I)$-rheology qualitatively reproduces the effect of the erodible bed on granular flow dynamics and deposits, such as the increase of runout distance with increasing thickness of the erodible bed. We show that the use of a constant friction coefficient in the multilayer model leads to the opposite behaviour. This multilayer model captures the strong change in shape of the velocity profile (from S-shaped to Bagnold-like) observed during the different phases of the highly transient flow, including the presence of static and flowing zones within the granular column.

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