On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part II — Application and Analysis

2020 ◽  
Author(s):  
Nina Wolfrum ◽  
Patrick Bechlars ◽  
Maximilian Beck ◽  
Christian Frey ◽  
Daniel Schlüß
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Meridional Flow ◽  
Computational Domain ◽  
Two Dimensional ◽  
Elementary Model ◽  
Nonreflecting Boundary Conditions ◽  
First Case ◽  
Geometrical Features ◽  
Rotational Surfaces

Abstract The flow in turbomachinery components is complex due to the relative motion of rotating and non-rotating elements. A proper design and prediction of physical phenomena requires reliable CFD tools. One important aspect is the incorporation of sophisticated algorithms at the boundaries of the computational domain. For inviscid, one-dimensional and two-dimensional Euler-flows there exist analytical solutions for the formulation of a boundary condition. Realistic applications, however, are viscous and consist of a complex three-dimensional character. Nevertheless, the analytical 2D nonreflecting boundary conditions are commonly used in CFD codes for their high computational efficiency and numerical robustness. The application becomes more challenging when the boundaries are close to geometrical features such as blades and vanes. In practical applications, the position of the boundaries is dictated by geometrical constraints and hence the proximity to the blading cannot always be avoided. The interaction of rotating and non-rotating geometrical features in a turbomachine produces complex flow patterns that propagate in the form of acoustic, vorticity and entropy waves. A boundary condition must be implemented in such a way that waves can propagate undisturbed out of the computational domain. Any reflection may unphysically affect the solution within the computational domain which is especially harmful to sensitive values such as unsteady aeroelastic quantities. But also steady-state computations may suffer from errors produced by reflective boundary conditions. The following paper is the second of two papers on the formulation of unsteady boundary conditions based on a two-dimensional analytical approach. The first part of this paper [6] explains how to extend 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. Two different formulations are discussed referring to the orientation of said rotational surfaces. In the first case the surfaces are oriented perpendicular to the boundary panel. In the second case the surfaces are aligned with the circumferentially averaged meridional flow velocity. In the present paper a thorough analysis of the two different approaches will be given. Both formulations of the boundary algorithm are validated on the basis of several elementary model flows. The behavior is analyzed for various unsteady wave patterns of different propagation directions with respect to the boundary. It will be shown that the alignment of the rotational surfaces with the meridional flow has a beneficial effect on the reflective behavior for the majority of the investigated flow conditions. The boundary conditions are then tested on realistic turbomachinery components in order to analyze their applicability on complex flows.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part I — Theory and Implementation

2020 ◽  
Author(s):  
Christian Frey ◽  
Daniel Schlüß ◽  
Nina Wolfrum ◽  
Patrick Bechlars ◽  
Maximilian Beck
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Radial Component ◽  
Rotational Surface ◽  
Nonreflecting Boundary Conditions ◽  
Meridional Velocity ◽  
The Impact ◽  
Rotational Surfaces

Abstract With unsteady flow simulations of industrial turbomachinery configurations becoming more and more affordable there is a growing need for accurate inlet and outlet boundary conditions as numerical reflections alone can lead to incorrect trends in engine efficiency, noise and aeroelastic analysis parameters. This is the first of two papers on the formulation of unsteady boundary conditions which have been implemented for both time-domain and frequency-domain solvers. Giles’ original idea for steady solvers to formulate the boundary condition in terms of characteristics generalizes to frequency-domain solvers. The boundary condition drives the value of the incoming characteristics to ideal values that are computed using the modal decomposition of linearized 2D Euler flows. The present paper explains how to generalize 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. For a flow with zero radial component and an annular boundary that is perpendicular to the machine axis, these surfaces are the cylindrical streamsurfaces. For more general flows and geometries, however, there is no natural choice for the rotational surfaces. In this paper, two choices are discussed: the surfaces that are generated by the boundary normals and those that are defined by the circumferentially averaged meridional velocity. The impact of the boundary condition on the stability of the harmonic-balance solver is analyzed by studying the pseudo-time evolution of certain energy integrals. For a model problem which consists of a small disturbance of an inviscid flow, the increase or decrease of this energy integral is shown to be directly related to the normal characteristic variables along the boundary. This shows that the actual boundary condition should be formulated as a control problem for the normal characteristics. Moreover, the application of the harmonic balance solver to a simple duct configuration with prescribed disturbances demonstrates that using the characteristics based on the meridional velocity may prevent the solver from converging. In contrast, the 2D theory can be formulated in a different surface without impairing the robustness of the overall approach. These findings are illustrated by a simple test case. The impact of the choice of the rotational surface for the 2D theory is studied for various duct segments and a low-pressure turbine configuration in the second paper. There it is shown that applying the 2D theory to the meridional-velocity surfaces may be advantageous in that it leads to more accurate results.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

Numerical Simulation of Nonlinear Wave Diffraction by a Vertical Cylinder in a Narrow Flume, Wide Tank, and Infinitely Wide Tank

2003 ◽  
Author(s):  
Alaa M. Mansour ◽  
A. Neil Williams
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Wave Diffraction ◽  
Open Boundary ◽  
Computational Domain ◽  
Open Boundary Condition ◽  
Lateral Direction ◽  
Cylinder Diameter ◽  
Side Walls

In this paper, a three dimensional numerical wave tank model has been used to simulate fully nonlinear wave diffraction by a uniform vertical circular cylinder. The cylinder has been placed in a narrow flume of a width equal to four times the cylinder diameter. The runup and the hydrodynamic forces on the cylinder has been compared to the results when a similar cylinder is placed in a similar tank but with a width equal to sixteen times the cylinder diameter. The model has been further extended by applying an open boundary condition to the side-walls to simulate an infinitely wide tank and hence more realistically simulate open sea condition. The proposed open boundary condition in the lateral direction is based on coupling of two prescribed boundary conditions, namely, numerical beach and Orlanski boundary conditions. The use of this coupled boundary condition has been found to be very efficient in eliminating any significant wave reflection from the side-walls back into the computational domain.

Gas Turbine Temperature Prediction Using Unsteady CFD and Realistic Non-Uniform 2D Combustor Exit Properties

2008 ◽  
Author(s):  
Fre´de´ric N. Felten ◽  
Semir Kapetanovic ◽  
D. Graham Holmes ◽  
Michael Ostrowski
Keyword(s):  
Boundary Conditions ◽  
Gas Turbine ◽  
Turbine Blades ◽  
Computational Cost ◽  
Computational Domain ◽  
Two Dimensional ◽  
Gas Temperature ◽  
Sliding Mesh ◽  
Flow Gradients ◽  
The Impact

Typical Computational Fluid Dynamics (CFD) studies performed on High Pressure Turbines (HPT) do not include the combustor domain in their analyses. Boundary conditions from the combustor exit have to be prescribed at the inlet of the computational domain for the first HPT nozzle. It is desirable to include the effect of combustor non-uniformities and flow gradients in order to enhance the accuracy of the aerodynamics and heat transfer predictions on the nozzle guide vanes and downstream turbine blades. The present work is the continuation of steady and quasi-unsteady studies performed previously by the authors. A fully unsteady nonlinear approach, also referred to as sliding mesh, is now used to investigate a first HPT stage and the impact of realistic non-uniformities and flow gradients found along the exit plane of a gas turbine combustor. Two Turbine Inlet Boundary Conditions (TIBC) are investigated. Simulations using a two-dimensional TIBC dependant on both the radial and circumferential directions are performed and compared to baseline analyses, where the previous two-dimensional TIBC is circumferentially averaged in order to generate inlet boundary conditions dependant only on the radial direction. The two elements included in the present work, combustor pitchwise non-uniformities and full unsteady blade row interactions are shown to: (1) alter the gas temperature profile predictions up to ±5%; (2) modify the surface temperature predictions by ±8% near the trailing edge of the vane suction side; (3) increase the overall pressure losses by roughly 1%, and (4) modified the ingestion behavior of the purge cavity flow. In addition, keeping in mind the tradeoff between improved predictions and computational cost, the use of an unsteady sliding mesh formulation, instead of a quasiunsteady frozen gust, reveals the importance of the two-way unsteady coupling between adjacent blade rows for temperature and pressure predictions.

scholarly journals Properties of RG interfaces for 2D boundary flows

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Anatoly Konechny
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Variational Method ◽  
Boundary Conditions ◽  
The Other ◽  
Two Dimensional ◽  
Conformal Boundary ◽  
First Approximation ◽  
Local Operators

Abstract We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J. Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D. Gaiotto for bulk 𝜙1,3 flows.

A Novel Mixing Plane Method Using Nonreflecting Boundary Conditions for Multirow Analysis in Turbomachines

2016 ◽  
Vol 138 (7) ◽  
Author(s):  
Fernando Gisbert ◽  
Roque Corral
Keyword(s):  
Boundary Condition ◽  
Steady State ◽  
Boundary Conditions ◽  
Reverse Flow ◽  
Plane Boundary ◽  
Normal Velocity ◽  
Nonreflecting Boundary Conditions ◽  
Plane Normal ◽  
Convergence Problems ◽  
New Formulation

A new formulation of the mixing plane boundary condition to analyze the steady-state interaction between adjacent rows of a turbomachine, used in conjunction with steady two-dimensional nonreflecting boundary conditions, is presented. Existing mixing plane formulations rely on the differences between some variables at the interface of adjacent rows to determine the boundary condition. These differences are driven to zero as the case is converged to the steady state. By contrast, the proposed approach determines the differences that result in the conservation of mass, momentum, and energy after the boundary condition is enforced, ensuring conservation at any instant during the iterative process. The reverse flow within the mixing plane boundary is naturally treated, but both inlet and outlet boundary conditions fail when the mixing plane normal velocity tends to zero, giving rise to sharp variations of the fluid variables that must be properly limited to prevent convergence problems. Some examples will be given to demonstrate the ability of the new method to resolve these cases while preserving the boundary condition robustness.

scholarly journals A framework for computing effective boundary conditions at the interface between free fluid and a porous medium

2017 ◽  
Vol 812 ◽  
pp. 866-889 ◽  
Author(s):  
Uǧis Lācis ◽  
Shervin Bagheri
Keyword(s):  
Boundary Condition ◽  
Porous Medium ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Accurate Method ◽  
Free Fluid ◽  
Computational Domain ◽  
Porous Bed ◽  
Effective Boundary ◽  
Effective Boundary Conditions

Interfacial boundary conditions determined from empirical or ad hoc models remain the standard approach to model fluid flows over porous media, even in situations where the topology of the porous medium is known. We propose a non-empirical and accurate method to compute the effective boundary conditions at the interface between a porous surface and an overlying flow. Using a multiscale expansion (homogenization) approach, we derive a tensorial generalized version of the empirical condition suggested by Beavers & Joseph (J. Fluid Mech., vol. 30 (01), 1967, pp. 197–207). The components of the tensors determining the effective slip velocity at the interface are obtained by solving a set of Stokes equations in a small computational domain near the interface containing both free flow and porous medium. Using the lid-driven cavity flow with a porous bed, we demonstrate that the derived boundary condition is accurate and robust by comparing an effective model to direct numerical simulations. Finally, we provide an open source code that solves the microscale problems and computes the velocity boundary condition without free parameters over any porous bed.

Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R2

2011 ◽  
Vol 10 (5) ◽  
pp. 1161-1183 ◽  
Author(s):  
Houde Han ◽  
Zhiwen Zhang
Keyword(s):  
Boundary Conditions ◽  
Gordon Equation ◽  
Splitting Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Computational Domain ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions

AbstractIn this paper the numerical solution of the two-dimensional sine-Gordon equation is studied. Split local artificial boundary conditions are obtained by the operator splitting method. Then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method, and some interesting propagation and collision behaviors of the solitary wave solutions are observed.

scholarly journals Classifying boundary conditions in JT gravity: from energy-branes to α-branes

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Akash Goel ◽  
Luca V. Iliesiu ◽  
Jorrit Kruthoff ◽  
Zhenbin Yang
Keyword(s):  
Boundary Condition ◽  
Black Holes ◽  
Boundary Conditions ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Ensemble Averaging ◽  
Matrix Integral ◽  
Two Dimensional Model ◽  
Definition Of ◽  
Extremal Black Holes

Abstract We classify the possible boundary conditions in JT gravity and discuss their exact quantization. Each boundary condition that we study will reveal new features in JT gravity related to its matrix integral interpretation, its factorization properties and ensemble averaging interpretation, the definition of the theory at finite cutoff, its relation to the physics of near-extremal black holes and, finally, its role as a two-dimensional model of cosmology.

Sign in / Sign up

Export Citation Format

Share Document