Tii£ RANDOM voKrEX METHOD for two-dimensional, nonstationary fl ows of a viscous li quid in a spa... more Tii£ RANDOM voKrEX METHOD for two-dimensional, nonstationary fl ows of a viscous li quid in a spatially periodic, iniinit e system of airfoils is considered. The main idea is to approximate the evolution of the vorticity by a large set of small "vortex particles" (vortex blobs), which are transported in the velocity field (convection) and perform random walks according to Wiener process with standard deviation depending on the viscosity {diffu sion). The velocity field is due to the induction of vortex blobs and includes also certain potentia l components. Since the fl ow domain is not simply connected, additional constraints concerning the vort icity production on the boundaries are introduced. They are necessary to obtain a solution with physically correct , single-valued pressure fie ld. The results of numerical calculations are al so presented.
2. Wybór konfiguracji geometrycznej ścianek kanału prowadzącej do destabilizacji przepływu.......... more 2. Wybór konfiguracji geometrycznej ścianek kanału prowadzącej do destabilizacji przepływu.............................................................................................................................. 6 3. Analiza teoretyczna i numeryczna wpływu orientacji kierunku pofalowania względem gradientu ciśnienia............................................................................................. 11 4. Analiza wpływu ścian bocznych na opory przepływu........................................................ 12 5. Wpływ ścian bocznych na stateczność przepływu w pofalowanym kanale........................ 13 6. Badanie nieliniowej saturacji zaburzenia przy pomocy modelu niskowymiarowego......... 14 7. Lagranżowska symulacja procesów mieszania................................................................... 16 8. Analiza struktury przepływu w geometrii uproszczonej..................................................... 21 9. Symulacja DNS przepływu cieczy w kanale z poprze...
This paper concerns the Lattice Boltzmann Method (LBM) which has recently become alternative, com... more This paper concerns the Lattice Boltzmann Method (LBM) which has recently become alternative, computationally efficient tool to simulate a Newtonian fluid flow, especially in complex geometries. A series of benchmark two-dimensional simulations with the LBM are presented and analyzed in order to assess the impact of the LBM grid refinement on the results of the simulations. The considered cases include stationary and time-dependent flows past a circular or a 45-degree rotated square cylinder confined in the channel with parallel walls. Next, the same flow configurations are investigated by means of the Spectral Element Method (SEM) and obtained results are used to cross-validate both approaches. The sensitivity to the resolution of representation of the obstacle’s shape in the LBM simulations is of the main interest. Particular attention is paid also to the effects of compressibility of the flow. Additionally, for unsteady flows the transient flow characteristics are compared in det...
Nowadays, cardiac disease is one of the most common cause of death. Each year almost one million ... more Nowadays, cardiac disease is one of the most common cause of death. Each year almost one million of angioplasty interventions and stents implantations are made all over the world. Unfortunately, in 20-30% of cases neointimal proliferations leads to restenosis occurring within the following period of 3-6 months. Three major factors are believed to contribute mostly to the edge restenosis: (a) mechanical damage of the artery-s wall caused by the stent implantation, (b) interaction between the stent and the blood constituents and (c) endothelial growth stimulation by small (lower that 1.5 Pa) and oscillating wall shear stress. Assuming that this last actor is particularly important, a numerical model of restenosis basing on wall shear stress distribution in the stented artery was elaborated. A numerical simulations of the development of in-stent restenosis have been performed and realistic geometric patterns of a progressing lumen reduction have been obtained
Summary Problem of mixing enhancement a viscous incompressible flow in a channel with wavy walls ... more Summary Problem of mixing enhancement a viscous incompressible flow in a channel with wavy walls is investigated theoretically and numerically. The wall waviness is unidirectional and it is oriented spanwise, i.e., the lines of constant elevation are parallel to the driving pressure gradient. It is shown that appropriately chosen wall waviness leads to destabilization at surprisingly low Reynolds numbers. The linear stability analysis shows that the critical Reynolds number Re cr can be reduced even under 60, i.e., by two orders of magnitude when compared to the Poiseuille flow between flat parallel planes. The unstable mode of disturbances has the form of a vortex array, which travels downstream. The remarkable feature is that the most destabilizing waviness does not introduce any additional flow resistance. The results of the stability analysis are consistent with the result of direct numerical simulation performed using the finite volume CFD package Fluent. Problem of mixing enha...
Journal of Theoretical and Applied Mechanics, 1995
The Karhunen-Loeve expansion, known in fluid dynamics as the Proper Orthogonal Decomposition (POD... more The Karhunen-Loeve expansion, known in fluid dynamics as the Proper Orthogonal Decomposition (POD), of random vector Melds is determined. These fields are the velocity of nonstationary flow past the cylindrical contour and uniform at infinity. The velocity field has been calculated with the Vortex Blobs Method. The POD procedure is carried out on the assumption of ergodicity which leads to an algebraic eigenvalue problem.
The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in inte... more The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in internal domains with many inlet/outlet sections. The novel variants of dissipative boundary conditions augmented by the inertia terms are used at the inlets/outlets of the flow domain. Volumetric flow rates or inlet/outlet average pressure are imposed as additional constrains imposed on a fluid motion. The variational formulations of the Stokes problem with such conditions and constrains are presented and the solution methods are proposed. These methods are based on superposition of appropriately defined auxiliary Stokes problems. Extension of the proposed methodology to the Navier-Stokes flows, based on the Operator-Integration-Factor-Splitting (OIFS) technique, is also described. Next, a nonlinear extension of the inertial-dissipative conditions is formulated and incorporated in the solution framework for the Navier-Stokes flows. Finally, an alternative approach to flow problems with such...
In this chapter, the application of the Gaussian regression models in the robust design and uncer... more In this chapter, the application of the Gaussian regression models in the robust design and uncertainty quantification is demonstrated. The computationally effective approach based on the Kriging method and relative expected improvement concept is described in detail. The new sampling criterion is proposed which leads to localization of the robust optimum in a limited number of steps. The methodology is employed to the optimal design process of the intake channel of the small turboprop engine.
The paper presents a mathematical formulation of the Lagrangian me-thod suitable for numerical si... more The paper presents a mathematical formulation of the Lagrangian me-thod suitable for numerical simulation of 3D viscous incompressible flows. The vorticity field is approximated by a large ensemble of vor-tex particles which move with the fluid (advection) and perform random walks (diffusion). The charges of the particles change with time due to the stretching term in the governing equation. The construction of the vortex particles ensures that the approximated vorticity field is stric-tly divergence-free at any time instant. The boundary condition at the surface of an immersed body is satisfied by the creation of new vortex particles near the surface. Various properties of induced velocity and vorticity fields are also discussed. Key words: vortex methods, vortex stretching, Fokker-Planck-Kolmogorow equation, Ito ̂ stochastic differential equations
Journal of Theoretical and Applied Mechanics, 2004
The paper presents a mathematical formulation of the Lagrangian method suitable for numerical sim... more The paper presents a mathematical formulation of the Lagrangian method suitable for numerical simulation of 3D viscous incompressible flows. The vorticity field is approximated by a large ensemble of vortex particles which move with the fluid (advection) and perform random walks (diffusion). The charges of the particles change with time due to the stretching term in the governing equation. The construction of the vortex particles ensures that the approximated vorticity field is strictly divergence-free at any time instant. The boundary condition at the surface of an immersed body is satisfied by the creation of new vortex particles near the surface. Various properties of induced velocity and vorticity fields are also discussed.
Journal of Theoretical and Applied Mechanics, 1993
In the paper the boundary problem of the Neumann type for the Laplace equation in a certain perio... more In the paper the boundary problem of the Neumann type for the Laplace equation in a certain periodic, 2-dimensional region is considered. The approach based on the boundary integral equation of the second kind has been suggested and the periodic version of such an equation has been obtained. The algorithm for solving this equation has been proposed and some sample results of numerical calculations have been presented.
An efficient numerical method for the linear stability equations of a spatially periodic channel ... more An efficient numerical method for the linear stability equations of a spatially periodic channel flow is presented. The method is based on global Fourier-Chebyshev approximation of a disturbance velocity field. The physical flow domain is embedded in a larger computational domain and the boundary conditions are re-formulated as internal conditions imposed at immersed boundaries. The advantage of this approach is an avoidance of domain mapping, leading to tremendous complication of governing equations and to excessive computational cost. The results of numerical tests are presented. Favorable convergence properties with respect to the length of the Fourier expansions are demonstrated.
The numerical implemcntation of the Lagrangian method using particles of the magnetization field ... more The numerical implemcntation of the Lagrangian method using particles of the magnetization field (magnetons) has been considered. A detailed description of essential elements of the algorithm has been provided. The presentation has focused on computations of stretching, where a novel integral-based rather than point wise approach has been proposed. The results of test computations, carried out for viscous flows past 2D and 3D bodies, have been presented. Difficulties with obtaining stable large-time simulations have been encountered and discussed. It has also been shown that, in contrast to flows around solid bodies, the vortex dynamics in the absence of boundaries can be successfully simulated. However, some consistent remeshing technique niay be necessary to achieve appropriate resolution.
Tii£ RANDOM voKrEX METHOD for two-dimensional, nonstationary fl ows of a viscous li quid in a spa... more Tii£ RANDOM voKrEX METHOD for two-dimensional, nonstationary fl ows of a viscous li quid in a spatially periodic, iniinit e system of airfoils is considered. The main idea is to approximate the evolution of the vorticity by a large set of small "vortex particles" (vortex blobs), which are transported in the velocity field (convection) and perform random walks according to Wiener process with standard deviation depending on the viscosity {diffu sion). The velocity field is due to the induction of vortex blobs and includes also certain potentia l components. Since the fl ow domain is not simply connected, additional constraints concerning the vort icity production on the boundaries are introduced. They are necessary to obtain a solution with physically correct , single-valued pressure fie ld. The results of numerical calculations are al so presented.
2. Wybór konfiguracji geometrycznej ścianek kanału prowadzącej do destabilizacji przepływu.......... more 2. Wybór konfiguracji geometrycznej ścianek kanału prowadzącej do destabilizacji przepływu.............................................................................................................................. 6 3. Analiza teoretyczna i numeryczna wpływu orientacji kierunku pofalowania względem gradientu ciśnienia............................................................................................. 11 4. Analiza wpływu ścian bocznych na opory przepływu........................................................ 12 5. Wpływ ścian bocznych na stateczność przepływu w pofalowanym kanale........................ 13 6. Badanie nieliniowej saturacji zaburzenia przy pomocy modelu niskowymiarowego......... 14 7. Lagranżowska symulacja procesów mieszania................................................................... 16 8. Analiza struktury przepływu w geometrii uproszczonej..................................................... 21 9. Symulacja DNS przepływu cieczy w kanale z poprze...
This paper concerns the Lattice Boltzmann Method (LBM) which has recently become alternative, com... more This paper concerns the Lattice Boltzmann Method (LBM) which has recently become alternative, computationally efficient tool to simulate a Newtonian fluid flow, especially in complex geometries. A series of benchmark two-dimensional simulations with the LBM are presented and analyzed in order to assess the impact of the LBM grid refinement on the results of the simulations. The considered cases include stationary and time-dependent flows past a circular or a 45-degree rotated square cylinder confined in the channel with parallel walls. Next, the same flow configurations are investigated by means of the Spectral Element Method (SEM) and obtained results are used to cross-validate both approaches. The sensitivity to the resolution of representation of the obstacle’s shape in the LBM simulations is of the main interest. Particular attention is paid also to the effects of compressibility of the flow. Additionally, for unsteady flows the transient flow characteristics are compared in det...
Nowadays, cardiac disease is one of the most common cause of death. Each year almost one million ... more Nowadays, cardiac disease is one of the most common cause of death. Each year almost one million of angioplasty interventions and stents implantations are made all over the world. Unfortunately, in 20-30% of cases neointimal proliferations leads to restenosis occurring within the following period of 3-6 months. Three major factors are believed to contribute mostly to the edge restenosis: (a) mechanical damage of the artery-s wall caused by the stent implantation, (b) interaction between the stent and the blood constituents and (c) endothelial growth stimulation by small (lower that 1.5 Pa) and oscillating wall shear stress. Assuming that this last actor is particularly important, a numerical model of restenosis basing on wall shear stress distribution in the stented artery was elaborated. A numerical simulations of the development of in-stent restenosis have been performed and realistic geometric patterns of a progressing lumen reduction have been obtained
Summary Problem of mixing enhancement a viscous incompressible flow in a channel with wavy walls ... more Summary Problem of mixing enhancement a viscous incompressible flow in a channel with wavy walls is investigated theoretically and numerically. The wall waviness is unidirectional and it is oriented spanwise, i.e., the lines of constant elevation are parallel to the driving pressure gradient. It is shown that appropriately chosen wall waviness leads to destabilization at surprisingly low Reynolds numbers. The linear stability analysis shows that the critical Reynolds number Re cr can be reduced even under 60, i.e., by two orders of magnitude when compared to the Poiseuille flow between flat parallel planes. The unstable mode of disturbances has the form of a vortex array, which travels downstream. The remarkable feature is that the most destabilizing waviness does not introduce any additional flow resistance. The results of the stability analysis are consistent with the result of direct numerical simulation performed using the finite volume CFD package Fluent. Problem of mixing enha...
Journal of Theoretical and Applied Mechanics, 1995
The Karhunen-Loeve expansion, known in fluid dynamics as the Proper Orthogonal Decomposition (POD... more The Karhunen-Loeve expansion, known in fluid dynamics as the Proper Orthogonal Decomposition (POD), of random vector Melds is determined. These fields are the velocity of nonstationary flow past the cylindrical contour and uniform at infinity. The velocity field has been calculated with the Vortex Blobs Method. The POD procedure is carried out on the assumption of ergodicity which leads to an algebraic eigenvalue problem.
The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in inte... more The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in internal domains with many inlet/outlet sections. The novel variants of dissipative boundary conditions augmented by the inertia terms are used at the inlets/outlets of the flow domain. Volumetric flow rates or inlet/outlet average pressure are imposed as additional constrains imposed on a fluid motion. The variational formulations of the Stokes problem with such conditions and constrains are presented and the solution methods are proposed. These methods are based on superposition of appropriately defined auxiliary Stokes problems. Extension of the proposed methodology to the Navier-Stokes flows, based on the Operator-Integration-Factor-Splitting (OIFS) technique, is also described. Next, a nonlinear extension of the inertial-dissipative conditions is formulated and incorporated in the solution framework for the Navier-Stokes flows. Finally, an alternative approach to flow problems with such...
In this chapter, the application of the Gaussian regression models in the robust design and uncer... more In this chapter, the application of the Gaussian regression models in the robust design and uncertainty quantification is demonstrated. The computationally effective approach based on the Kriging method and relative expected improvement concept is described in detail. The new sampling criterion is proposed which leads to localization of the robust optimum in a limited number of steps. The methodology is employed to the optimal design process of the intake channel of the small turboprop engine.
The paper presents a mathematical formulation of the Lagrangian me-thod suitable for numerical si... more The paper presents a mathematical formulation of the Lagrangian me-thod suitable for numerical simulation of 3D viscous incompressible flows. The vorticity field is approximated by a large ensemble of vor-tex particles which move with the fluid (advection) and perform random walks (diffusion). The charges of the particles change with time due to the stretching term in the governing equation. The construction of the vortex particles ensures that the approximated vorticity field is stric-tly divergence-free at any time instant. The boundary condition at the surface of an immersed body is satisfied by the creation of new vortex particles near the surface. Various properties of induced velocity and vorticity fields are also discussed. Key words: vortex methods, vortex stretching, Fokker-Planck-Kolmogorow equation, Ito ̂ stochastic differential equations
Journal of Theoretical and Applied Mechanics, 2004
The paper presents a mathematical formulation of the Lagrangian method suitable for numerical sim... more The paper presents a mathematical formulation of the Lagrangian method suitable for numerical simulation of 3D viscous incompressible flows. The vorticity field is approximated by a large ensemble of vortex particles which move with the fluid (advection) and perform random walks (diffusion). The charges of the particles change with time due to the stretching term in the governing equation. The construction of the vortex particles ensures that the approximated vorticity field is strictly divergence-free at any time instant. The boundary condition at the surface of an immersed body is satisfied by the creation of new vortex particles near the surface. Various properties of induced velocity and vorticity fields are also discussed.
Journal of Theoretical and Applied Mechanics, 1993
In the paper the boundary problem of the Neumann type for the Laplace equation in a certain perio... more In the paper the boundary problem of the Neumann type for the Laplace equation in a certain periodic, 2-dimensional region is considered. The approach based on the boundary integral equation of the second kind has been suggested and the periodic version of such an equation has been obtained. The algorithm for solving this equation has been proposed and some sample results of numerical calculations have been presented.
An efficient numerical method for the linear stability equations of a spatially periodic channel ... more An efficient numerical method for the linear stability equations of a spatially periodic channel flow is presented. The method is based on global Fourier-Chebyshev approximation of a disturbance velocity field. The physical flow domain is embedded in a larger computational domain and the boundary conditions are re-formulated as internal conditions imposed at immersed boundaries. The advantage of this approach is an avoidance of domain mapping, leading to tremendous complication of governing equations and to excessive computational cost. The results of numerical tests are presented. Favorable convergence properties with respect to the length of the Fourier expansions are demonstrated.
The numerical implemcntation of the Lagrangian method using particles of the magnetization field ... more The numerical implemcntation of the Lagrangian method using particles of the magnetization field (magnetons) has been considered. A detailed description of essential elements of the algorithm has been provided. The presentation has focused on computations of stretching, where a novel integral-based rather than point wise approach has been proposed. The results of test computations, carried out for viscous flows past 2D and 3D bodies, have been presented. Difficulties with obtaining stable large-time simulations have been encountered and discussed. It has also been shown that, in contrast to flows around solid bodies, the vortex dynamics in the absence of boundaries can be successfully simulated. However, some consistent remeshing technique niay be necessary to achieve appropriate resolution.
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