ACOUSTIC PERFORMANCE OF NONREFLECTING BOUNDARY CONDITIONS FOR A RANGE OF INCIDENT ANGLES

2008 ◽  
Vol 16 (01) ◽  
pp. 11-29 ◽  
Author(s):  
M. MESBAH ◽  
J. MEYERS ◽  
M. BAELMANS
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Incident Angle ◽  
Test Cases ◽  
Acoustic Performance ◽  
Nonreflecting Boundary Conditions ◽  
Selection Of

A detailed study on the accuracy and reflection behavior of nonreflection boundary conditions is presented. To this end, a selection of five distinct nonreflecting boundary conditions is evaluated and the mechanisms which dominate the reflection properties of boundary conditions are identified based on a rigorous quantification of reflection levels. It is shown that the reflection behavior is significantly affected by the incident angle. The relation between the boundary condition effectiveness and the incident angle is investigated and the reflection rates as a function of the incident angle are quantified. Furthermore, the obtained results are used to predict the reflections from boundary conditions in general applications. It is shown that these predictions are reliable for different test cases.

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.

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.

Analysis of a Fixed-Guided Compliant Beam With an Inflection Point Using the Pseudo-Rigid-Body Model (PRBM) Concept

2012 ◽  
Author(s):  
Ashok Midha ◽  
Sushrut G. Bapat ◽  
Adarsh Mavanthoor ◽  
Vivekananda Chinta
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Rigid Body ◽  
Inflection Point ◽  
Equilibrium Equations ◽  
Displacement Boundary ◽  
Body Model ◽  
Rigid Body Model ◽  
Domain Concept ◽  
Selection Of

This paper provides an efficient method of analysis for a fixed-guided compliant beam with an inflection point, subjected to beam end load or displacement boundary conditions, or a combination thereof. To enable this, such a beam is modeled as a pair of well-established pseudo-rigid-body models (PRBMs) for fixed-free compliant beam segments. The analysis procedure relies on the properties of inflection in developing the necessary set of static equilibrium equations for solution. The paper further discusses the multiplicity of possible solutions, including displacement configurations, for any two specified beam end boundary conditions, depending on the locations of the effecting force and/or displacement boundary conditions. A unique solution may exist when a third beam end boundary condition is specified; however, this selection is not unconditional. A deflection domain concept is proposed to assist with the selection of the third boundary condition in a more realistic manner.

A SLOPING BOUNDARY CONDITION FOR EFFICIENT PE CALCULATIONS IN RANGE–DEPENDENT ACOUSTIC MEDIA

1996 ◽  
Vol 04 (01) ◽  
pp. 11-27 ◽  
Author(s):  
GARY H. BROOKE ◽  
DAVID J. THOMSON ◽  
PHILIP M. WORT
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Boundary Conditions ◽  
Standard Test ◽  
Numerical Results ◽  
Normal Displacement ◽  
Test Cases ◽  
Approximate Form ◽  
Acoustic Media ◽  
Horizontal Interface

The traditional one-way parabolic equation (PE) formulation for range-dependent layered acoustic media is modified to include effects associated with the boundary conditions along a sloping interface. Essentially, the boundary condition for continuity of the normal displacement along a sloping interface is cast in an approximate form which does not depend on range but does contain terms up to second order in the derivatives with respect to depth. The new sloping-boundary condition is then applied along an "equivalent" horizontal interface within each range-independent step of the PE. Numerical results obtained for standard test cases indicate that the sloping-boundary condition, incorporated into a one-way PE, maintains the efficiency yet improves the accuracy of forward predictions.

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.

Analysis of a Fixed-Guided Compliant Beam With an Inflection Point Using the Pseudo-Rigid-Body Model Concept

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Ashok Midha ◽  
Sushrut G. Bapat ◽  
Adarsh Mavanthoor ◽  
Vivekananda Chinta
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Rigid Body ◽  
Inflection Point ◽  
Compliant Mechanism ◽  
Analysis Method ◽  
Model Concept ◽  
Displacement Boundary ◽  
Rigid Body Model ◽  
Selection Of

This paper provides an efficient method of analysis for a fixed-guided compliant beam with an inflection point, subjected to beam end load or displacement boundary conditions, or a combination thereof. To enable this, such a beam is modeled as a pair of well-established pseudo-rigid-body models (PRBMs) for fixed-free compliant beam segments. The analysis procedure relies on the properties of inflection in developing the necessary set of parametric, static equilibrium and compatibility equations for solution. The paper further discusses the multiplicity of possible solutions, including displacement configurations, for any two specified beam end displacement boundary conditions, depending on the locations and types of the effecting loads on the beam to meet these boundary conditions. A unique solution may exist when a third beam end displacement boundary condition is specified; however, this selection is not unconditional. A concept of characteristic deflection domain is proposed to assist with the selection of the third boundary condition to yield a realistic solution. The analysis method is also used to synthesize a simple, fully compliant mechanism utilizing the fixed-guided compliant segments.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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