**90**

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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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**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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**The hydrostatic approximation of compressible anisotropic Navier-Stokes equations**

Gao, H.; Nečasová, Šárka; Tang, T.

2022 - English
The aim of the paper is to give a rigorous derivation of the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in the general large scale geophysical motions meaning that the vertical scale is signi cantly smaller than horizontal.
Keywords:
*anisotropic Naiver-Stokes equations; aspect ratio limit; hydrostatic approximation*
Available in digital repository of the ASCR
The hydrostatic approximation of compressible anisotropic Navier-Stokes equations

The aim of the paper is to give a rigorous derivation of the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio of the depth ...

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**Numerical tests of vanishing diffusion stabilization in Oldroyd-B fluid flow simulations**

Pires, M.; Bodnár, Tomáš

2021 - English
This work presents some numerical tests of finite element solution of incompressible Oldroyd-B fluids flows, using different types of numerical stabilization. In this study the diffusive term (Laplacian of extra stress) is added to the tensorial constitutive relation where it is multiplied by a coefficient, that is variable in time. The goal is to make this diffusion coefficient vanish in time, so that the final solution remains unaffected by the added diffusion term. A series of numerical tests was performed for the steady two-dimensional Oldroyd-B fluid flow in corrugated channel (tube) to compare different versions of the vanishing stabilization terms and assess their efficiency in enforcing the solution convergence, without affecting the final steady state.
Keywords:
*finite element method; oldroyd-B fluid; numerical stabilization; stress diffusion*
Available in digital repository of the ASCR
Numerical tests of vanishing diffusion stabilization in Oldroyd-B fluid flow simulations

This work presents some numerical tests of finite element solution of incompressible Oldroyd-B fluids flows, using different types of numerical stabilization. In this study the diffusive term ...

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**Second-order model for atmospheric turbulence without critical Richardson number**

Caggio, M.; Schiavon, M.; Tampieri, F.; Bodnár, Tomáš

2021 - English
The purpose of this communication is to present a derivation of the non-dimensional vertical gradients of the mean wind speed and mean potential temperature expressed in terms of the so-called similarity functions for very stable conditions of the atmosphere where theoretical approaches provide conflicting results (see e.g. Luhar et al. [19]). The result is based on the analysis of the second-order model equations in the boundary layer approximations in which new heat flux equations are proposed. The model employs a recent closure for the pressure-temperature correlation, avoiding the issue of a critical treshold for the Richardson number.
Keywords:
*atmospheric boundary layer; second-order closure model; turbulence parameterisations; strong strati cation*
Available in digital repository of the ASCR
Second-order model for atmospheric turbulence without critical Richardson number

The purpose of this communication is to present a derivation of the non-dimensional vertical gradients of the mean wind speed and mean potential temperature expressed in terms of the so-called ...

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**Note on the mathematical analysis of the motion of a rigid body in a generalized incompressible Navier-Stokes fluid**

Al Baba, H.; Ghosh, Amrita; Muha, B.; Nečasová, Šárka

2021 - English
Wedeal with a fluid-structure interaction problem: a motion of the rigid body inside a bounded domain filled by a fluid. We consider a viscous incompressible fluid described by the generalized incompressible Navier-Stokes equations which include cases of Newtonian and non-Newtonian f luids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and the motion of the solid is governed by Newton’s laws. We also investigate the case of the nonlinear slip condition.The main results assert the existence of strong solutions, in an Lp − Lq setting.
Keywords:
*Navier-Stokes equations; non-Newtonian fluid; Navier slip condition; strong solution*
Available in digital repository of the ASCR
Note on the mathematical analysis of the motion of a rigid body in a generalized incompressible Navier-Stokes fluid

Wedeal with a fluid-structure interaction problem: a motion of the rigid body inside a bounded domain filled by a fluid. We consider a viscous incompressible fluid described by the generalized ...

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**Excessive extrapolations of Einstein's equations**

Křížek, Michal; Somer, L.

2020 - English
The standard cosmological model is surprisingly quite thoroughly investigated even though it possesses many paradoxes. We present several arguments indicating why excessive extrapolations of Einstein's equations to cosmological distances are questionable. First, we show how to express explicitly the first of Einstein's 10 partial differential equations to demonstrate their extremely large complexity. Therefore, it would be very difficult to find their solution for two or more bodies to model, e.g., the evolution of the Solar system. Further, we present some unexpected failures of the Schwarzschild and Friedmann solution of these equations. Then we explain why application of Einstein's equations to the whole universe represents incorrect extrapolations that lead to dark matter, dark energy, and several unrealistic situations. Finally, we give 10 further arguments showing why celebrated Einstein's equations do not describe reality well.
Keywords:
*Einstein's equations; Schwarzschild solution; Friedmann equation*
Fulltext is available at external website.
Excessive extrapolations of Einstein's equations

The standard cosmological model is surprisingly quite thoroughly investigated even though it possesses many paradoxes. We present several arguments indicating why excessive extrapolations of ...

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**The effect of submeso motions on second-order moment budgets in the stable atmospheric boundary layer**

Schiavon, M.; Tampieri, F.; Caggio, M.; Bodnár, Tomáš

2020 - English
The effect of submeso motions on small-scale turbulence is studied considering the budget of the vertical flux of stream-wise momentum, ⟨uw⟩, in the atmospheric stable boundary layer (SBL). A parameter expressing the strength of the submeso effect is defined, and the budget is evaluated from observations both for small and large submeso effect. It results that submeso motions affect the efficiency of the vertical transport by small-scale turbulence, having implications on the terms composing the momentum flux budget and on its corresponding closures.
Keywords:
*submeso motions; pressure redistribution; second-order moment equations*
Available in digital repository of the ASCR
The effect of submeso motions on second-order moment budgets in the stable atmospheric boundary layer

The effect of submeso motions on small-scale turbulence is studied considering the budget of the vertical flux of stream-wise momentum, ⟨uw⟩, in the atmospheric stable boundary layer (SBL). A ...

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**On the influence of diffusion stabilization in Oldroyd-B fluid flow simulations**

Pires, M.; Bodnár, Tomáš

2020 - English
This work presents some numerical tests of finite element solution of incompressible Oldroyd-B fluid flows. The effect of numerical stabilization using artificial stress diffusion is investigated in detail. The limits of Weissenberg number We for which it is possible to obtain the numerical solution were studied depending on the Reynolds number Re and the diffusion parameter. Series of numerical tests were performed for steady two-dimensional Oldroyd-B fluid flow in corrugated channel (tube). The numerical results clearly proved the advantage (higher attainable We) of stabilized numerical method over the classical formulation without the artificial stress diffusion.
Keywords:
*finite element method; Oldroyd-B fluid; numerical stabilization; stres diffusion*
Available in digital repository of the ASCR
On the influence of diffusion stabilization in Oldroyd-B fluid flow simulations

This work presents some numerical tests of finite element solution of incompressible Oldroyd-B fluid flows. The effect of numerical stabilization using artificial stress diffusion is investigated in ...

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