**97**

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**On the development of a numerical model for the simulation of air flow in the human airways**

Lancmanová, Anna; Bodnár, Tomáš; Sequeira, A.

2023 - English
This contribution reports on an ongoing study focusing on reduced order models for incompressible viscous fluid flow in two dimensional channels. A finite difference solver was developed using a simple implementation of the immersed boundary method to represent the channel geometry. The solver was validated for unsteady flow by comparing the obtained two-dimensional numerical solutions with analytical profiles computed from the Womersley solution. Finally the 2D model was coupled to a simple 1D extension simulating the flow in axisymmetric elastic vessel (tube). Some of the coupling principles and implementation issues are discussed in detail.
Keywords:
*reduced order model; incompressible Navier-Stokes equations; finite difference approximation; coupling method*
Available in digital repository of the ASCR
On the development of a numerical model for the simulation of air flow in the human airways

This contribution reports on an ongoing study focusing on reduced order models for incompressible viscous fluid flow in two dimensional channels. A finite difference solver was developed using a ...

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**Spherical basis function approximation with particular trend functions**

Segeth, Karel

2023 - English
The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the hbox{$d$-dimensional} Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d=3$. We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula can be useful in the interpretation of many physical measurements. We show an example concerned with the measurement of anisotropy of magnetic susceptibility having extensive applications in geosciences and present numerical difficulties connected with the high condition number of the matrix of the system defining the interpolation.
Keywords:
*spherical interpolation; spherical radial basis function; inverse multiquadric*
Available in digital repository of the ASCR
Spherical basis function approximation with particular trend functions

The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d-1)$-dimensional unit sphere surface in the hbox{$d$-dimensional} Euclidean space and the spherical RBF ...

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**Validation of numerical simulations of a simple immersed boundary solver for fluid flow in branching channels**

Keslerová, R.; Lancmanová, Anna; Bodnár, Tomáš

2023 - English
This work deals with the flow of incompressible viscous fluids in a two-dimensional branching channel. Using the immersed boundary method, a new finite difference solver was developed to interpret the channel geometry. The numerical results obtained by this new solver are compared with the numerical simulations of the older finite volume method code and with the results obtained with OpenFOAM. The aim of this work is to verify whether the immersed boundary method is suitable for fluid flow in channels with more complex geometries with difficult grid generation.
Keywords:
*immersed boundary method; finite volume method; OpenFOAM*
Available in digital repository of the ASCR
Validation of numerical simulations of a simple immersed boundary solver for fluid flow in branching channels

This work deals with the flow of incompressible viscous fluids in a two-dimensional branching channel. Using the immersed boundary method, a new finite difference solver was developed to interpret the ...

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**On the problem of singular limit**

Caggio, Matteo; Ducomet, B.; Nečasová, Šárka; Tang, T.

2023 - English
We consider the problem of singular limit of the compressible Euler system confined to a straight layer Ωδ = (0, δ)×R², δ > 0. In the regime of low Mach number limit and reduction of dimension the convergence to the strong solution of the 2D incompressible Euler system is shown.
Keywords:
*compressible Euler equations; dissipative measure-valued solutions; low Mach number; thin domain*
Available in digital repository of the ASCR
On the problem of singular limit

We consider the problem of singular limit of the compressible Euler system confined to a straight layer Ωδ = (0, δ)×R², δ > 0. In the regime of low Mach number limit and reduction of dimension the ...

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**Interpolation with restrictions -- role of the boundary conditions and individual restrictions**

Valášek, Jan; Sváček, P.

2023 - English
The contribution deals with the remeshing procedure between two computational finite element meshes. The remeshing represented by the interpolation of an approximate solution onto a new mesh is needed in many applications like e.g. in aeroacoustics, here we are particularly interested in the numerical flow simulation of a gradual channel collapse connected with a~severe deterioration of the computational mesh quality. Since the classical Lagrangian projection from one mesh to another is a dissipative method not respecting conservation laws, a conservative interpolation method introducing constraints is described. The constraints have form of Lagrange multipliers enforcing conservation of desired flow quantities, like e.g. total fluid mass, flow kinetic energy or flow potential energy. Then the interpolation problem turns into an error minimization problem, such that the resulting quantities of proposed interpolation satisfy these physical properties while staying as close as possible to the results of Lagrangian interpolation in the L2 norm. The proposed interpolation scheme does not impose any restrictions on mesh generation process and it has a relatively low computational cost. The implementation details are discussed and test cases are shown.
Keywords:
*interpolation; Lagrange multiplier; Lagrange projection*
Available in digital repository of the ASCR
Interpolation with restrictions -- role of the boundary conditions and individual restrictions

The contribution deals with the remeshing procedure between two computational finite element meshes. The remeshing represented by the interpolation of an approximate solution onto a new mesh is needed ...

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**Hidden symmetry in turbulence and analytic study of shell models**

Caggio, Matteo

2023 - English
This short communication concerns symmetries in developed turbulence and analytic study of shell models. However scale-invariance is broken due to the intermittency phenomenon, is possible to established a hidden self-similarity in turbulent flows. Using a shell model, the author in [18] (see also [19]) addressed the problem deriving a scaling symmetry for the inviscid equations. Here, first we discuss the analysis presented in [18], then, from the mathematical perspective, we propose an analytic study for the shell model with the presence of the viscous terms. This brief paper should be understood as an introductory note to this new scaling symmetry with implications for mathematical analysis [5].
Keywords:
*turbulence; scale-invariance symmetry; intermittency; shell-models*
Available in digital repository of the ASCR
Hidden symmetry in turbulence and analytic study of shell models

This short communication concerns symmetries in developed turbulence and analytic study of shell models. However scale-invariance is broken due to the intermittency phenomenon, is possible to ...

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**100 years of the Friedmann equation**

Křížek, Michal

2022 - English
In 1922, Alexander Friedmann applied Einstein’s equations to a three-dimensional sphere to describe the evolution of our universe. In this way he obtained a nonlinear ordinary differential equation (called after him) for the expansion function representing the radius of that sphere. At present, the standard cosmological ΛCDM model of the universe is based just on the Friedmann equation. It needs a significant amount of dark matter, about six times that of the usual baryonic matter, besides an even larger amount of dark energy to be consistent with the real universe. But to date, both dark matter and dark energy have remained without concrete evidence based on direct physical measurements. We present several arguments showing that such a claimed amount of dark matter and dark energy can only be the result of vast overestimation, incorrect extrapolations, and that it does not correspond to the real universe. The spatial part of our universe seems to be locally flat and thus it can be locally modeled by the Euclidean space. However, Friedmann did not consider the flat space with zero curvature. Therefore, in the second part of this paper we will derive a general form of the corresponding metric tensor satisfying Einstein’s equations with zero right-hand side.
Keywords:
*Einstein's equations; modeling error; incorrect extrapolations; dark matter*
Fulltext is available at external website.
100 years of the Friedmann equation

In 1922, Alexander Friedmann applied Einstein’s equations to a three-dimensional sphere to describe the evolution of our universe. In this way he obtained a nonlinear ordinary differential equation ...

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**Cosmology on Small Scales 2022: Dark Energy and the Local Hubble Expansion Problem**

Křížek, Michal; Dumin, Y. V.

2022 - English
One hundred years ago, Russian mathematician and physicist Alexander A. Friedmann applied the system of Einstein equations to the three-dimensional sphere with a time varying radius. In this way, he obtained a nonlinear ordinary differential equation which is called the Friedmann equation after him and serves now as a cornerstone of the standard cosmological model. Unfortunately, it is well known that this model exhibits a number of paradoxes. Thus, the main goal of the CSS 2022 Conference Proceedings is to discuss whether and how the Friedmann equation can be applied at the various spatial scales, from our local cosmic neighborhood up to the whole Universe, and if the existence of dark matter and dark energy are merely artifacts of the excessive extrapolations.
Keywords:
*Friedmann equation; dark matter; dark energy; local Hubble expansion*
Fulltext is available at external website.
Cosmology on Small Scales 2022: Dark Energy and the Local Hubble Expansion Problem

One hundred years ago, Russian mathematician and physicist Alexander A. Friedmann applied the system of Einstein equations to the three-dimensional sphere with a time varying radius. In this way, he ...

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**Numerical validation of a simple immersed boundary solver for branched channels simulations**

Lancmanová, A.; Bodnár, Tomáš; Keslerová, D.

2022 - English
This contribution reports on an ongoing study of incompressible viscous fluid flow in two dimensional branched channels. A new finite difference solver was developed using a simple implementation of an immersed boundary method to represent the channel geometry. Numerical solutions obtained using this new solver are compared with outputs of an older finite volume code working on classical wall tted structured multiblock grid. Besides of the comparative evaluation of obtained solution, the aim is to verify whether the immersed boundary method is suitable (accurate and e cient enough) for simulations of flow in channels with complicated geometry where the the grid generation might be challenging.
Keywords:
*branching channel; incompressible Navier-Stokes; finite difference; artificial compressibility*
Available in digital repository of the ASCR
Numerical validation of a simple immersed boundary solver for branched channels simulations

This contribution reports on an ongoing study of incompressible viscous fluid flow in two dimensional branched channels. A new finite difference solver was developed using a simple implementation of ...

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**Numerical assessment of stratification influence in simple algebraic turbulence model**

Uhlíř, V.; Bodnár, Tomáš; Caggio, Matteo

2022 - English
This paper presents rst few results obtained using a newly developed test code aimed at validation and cross-comparison of turbulence models to be applied in environmental flows. A simple code based on nite di erence discretization is constructed to solve steady flows of incompresible non-homogeneous (variable denstity) fluids. For the rst tests a simple algebraic turbulence model was implemented, containing stability function depending on the stratification via the gradient Richardson number. Numerical tests were performed in order to explore the capabilities of the new code and to get some insight into its behavior under di erent stratification. The two-dimensional simulations were performed using immersed boundary method for the flow over low smooth hill. The resulting flow fields are compared for selected Richarson numbers ranging from stable up to unstable strati cation conditions.
Keywords:
*stratification; immersed boundary; algebraic turbulence model; artifcial compressibility method*
Available in digital repository of the ASCR
Numerical assessment of stratification influence in simple algebraic turbulence model

This paper presents rst few results obtained using a newly developed test code aimed at validation and cross-comparison of turbulence models to be applied in environmental flows. A simple code based ...

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