Theory and application of nonlinear wave propagation phenomena in combined reaction/separation processes

Reaction separation processes, reactive distillation, chromatographic reactor, equilibrium theory, nonlinear waves, process control, observer design, asymptoticaly exact input/output-linearization

Higher order finite elements and the fictitious domain concept for wave propagation analysis

Magdeburg, Univ., Fak. für Maschinenbau, Diss., 2014

A pseudo-spectral method for the simulation of poro-elastic seismic wave propagation in 2D polar coordinates using domain decomposition

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in 2D polar coordinates. An important application of this method and its extensions will be the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh, which can be arbitrarily heterogeneous, consisting of two or more concentric rings representing the fluid in the center and the surrounding porous medium. The spatial discretization is based on a Chebyshev expansion in the radial direction and a Fourier expansion in the azimuthal direction and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method is used to match the fluid-solid boundary conditions based on the method of characteristics. This multi-domain approach allows for significant reductions of the number of grid points in the azimuthal direction for the inner grid domain and thus for corresponding increases of the time step and enhancements of computational efficiency. The viability and accuracy of the proposed method has been rigorously tested and verified through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently bench-marked solution for 2D Cartesian coordinates. Finally, the proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is adequately handled.

Quantitative comparison of simulations of seismic wave propagation in heterogeneous poro-elastic media involving fluid-solid interfaces and in equivalent visco-elastic solids

There is increasing evidence to suggest that the presence of mesoscopic heterogeneities constitutes the predominant attenuation mechanism at seismic frequencies. As a consequence, centimeter-scale perturbations of the subsurface physical properties should be taken into account for seismic modeling whenever detailed and accurate responses of the target structures are desired. This is, however, computationally prohibitive since extremely small grid spacings would be necessary. A convenient way to circumvent this problem is to use an upscaling procedure to replace the heterogeneous porous media by equivalent visco-elastic solids. In this work, we solve Biot's equations of motion to perform numerical simulations of seismic wave propagation through porous media containing mesoscopic heterogeneities. We then use an upscaling procedure to replace the heterogeneous poro-elastic regions by homogeneous equivalent visco-elastic solids and repeat the simulations using visco-elastic equations of motion. We find that, despite the equivalent attenuation behavior of the heterogeneous poro-elastic medium and the equivalent visco-elastic solid, the seismograms may differ due to diverging boundary conditions at fluid-solid interfaces, where there exist additional options for the poro-elastic case. In particular, we observe that the seismograms agree for closed-pore boundary conditions, but differ significantly for open-pore boundary conditions. This is an interesting result, which has potentially important implications for wave-equation-based algorithms in exploration geophysics involving fluid-solid interfaces, such as, for example, wave field decomposition.

A pseudo-spectral method for simulating of poro-elastic seismic wave propagation in complex borehole environments

We present a novel numerical approach for the comprehensive, flexible, and accurate simulation of poro-elastic wave propagation in cylindrical coordinates. An important application of this method is the modeling of complex seismic wave phenomena in fluid-filled boreholes, which represents a major, and as of yet largely unresolved, computational problem in exploration geophysics. In view of this, we consider a numerical mesh consisting of three concentric domains representing the borehole fluid in the center, the borehole casing and the surrounding porous formation. The spatial discretization is based on a Chebyshev expansion in the radial direction, Fourier expansions in the other directions, and a Runge-Kutta integration scheme for the time evolution. A domain decomposition method based on the method of characteristics is used to match the boundary conditions at the fluid/porous-solid and porous-solid/porous-solid interfaces. The viability and accuracy of the proposed method has been tested and verified in 2D polar coordinates through comparisons with analytical solutions as well as with the results obtained with a corresponding, previously published, and independently benchmarked solution for 2D Cartesian coordinates. The proposed numerical solution also satisfies the reciprocity theorem, which indicates that the inherent singularity associated with the origin of the polar coordinate system is handled adequately.

Wave propagation in a medium with disordered excitability

The effect of quenched disorder on the propagation of autowaves in excitable media is studied both experimentally and numerically in relation to the light-sensitive Belousov-Zhabotinsky reaction. The spatial disorder is introduced through a random distribution with two different levels of transmittance. In one dimension the (time-averaged) wave speed is smaller than the corresponding to a homogeneous medium with the mean excitability. Contrarily, in two dimensions the velocity increases due to the roughening of the front. Results are interpreted using kinematic and scaling arguments. In particular, for d = 2 we verify a theoretical prediction of a power-law dependence for the relative change of the propagation speed on the disorder amplitude.

Regular wave propagation out of noise in chemical active media

A pacemaker, regularly emitting chemical waves, is created out of noise when an excitable photosensitive Belousov-Zhabotinsky medium, strictly unable to autonomously initiate autowaves, is forced with a spatiotemporal patterned random illumination. These experimental observations are also reproduced numerically by using a set of reaction-diffusion equations for an activator-inhibitor model, and further analytically interpreted in terms of genuine coupling effects arising from parametric fluctuations. Within the same framework we also address situations of noise-sustained propagation in subexcitable media.

Wave propagation in a medium with disordered excitability

The effect of quenched disorder on the propagation of autowaves in excitable media is studied both experimentally and numerically in relation to the light-sensitive Belousov-Zhabotinsky reaction. The spatial disorder is introduced through a random distribution with two different levels of transmittance. In one dimension the (time-averaged) wave speed is smaller than the corresponding to a homogeneous medium with the mean excitability. Contrarily, in two dimensions the velocity increases due to the roughening of the front. Results are interpreted using kinematic and scaling arguments. In particular, for d = 2 we verify a theoretical prediction of a power-law dependence for the relative change of the propagation speed on the disorder amplitude.

Regular wave propagation out of noise in chemical active media

A pacemaker, regularly emitting chemical waves, is created out of noise when an excitable photosensitive Belousov-Zhabotinsky medium, strictly unable to autonomously initiate autowaves, is forced with a spatiotemporal patterned random illumination. These experimental observations are also reproduced numerically by using a set of reaction-diffusion equations for an activator-inhibitor model, and further analytically interpreted in terms of genuine coupling effects arising from parametric fluctuations. Within the same framework we also address situations of noise-sustained propagation in subexcitable media.

Quantitative comparison between simulations of seismic wave propagation in heterogeneous poro-elastic media and equivalent visco-elastic solids for marine-type environments

There is increasing evidence to suggest that the presence of mesoscopic heterogeneities constitutes an important seismic attenuation mechanism in porous rocks. As a consequence, centimetre-scale perturbations of the rock physical properties should be taken into account for seismic modelling whenever detailed and accurate responses of specific target structures are desired, which is, however, computationally prohibitive. A convenient way to circumvent this problem is to use an upscaling procedure to replace each of the heterogeneous porous media composing the geological model by corresponding equivalent visco-elastic solids and to solve the visco-elastic equations of motion for the inferred equivalent model. While the overall qualitative validity of this procedure is well established, there are as of yet no quantitative analyses regarding the equivalence of the seismograms resulting from the original poro-elastic and the corresponding upscaled visco-elastic models. To address this issue, we compare poro-elastic and visco-elastic solutions for a range of marine-type models of increasing complexity. We found that despite the identical dispersion and attenuation behaviour of the heterogeneous poro-elastic and the equivalent visco-elastic media, the seismograms may differ substantially due to diverging boundary conditions, where there exist additional options for the poro-elastic case. In particular, we observe that at the fluid/porous-solid interface, the poro- and visco-elastic seismograms agree for closed-pore boundary conditions, but differ significantly for open-pore boundary conditions. This is an important result which has potentially far-reaching implications for wave-equation-based algorithms in exploration geophysics involving fluid/porous-solid interfaces, such as, for example, wavefield decomposition.

Comparison of Setting Time Measured Using Ultrasonic Wave Propagation with Saw-Cutting Times on Pavements, TR-675, 2015

At present, there is little fundamental guidance available to assist contractors in choosing when to schedule saw cuts on joints. To conduct pavement finishing and sawing activities effectively, however, contractors need to know when a concrete mixture is going to reach initial set, or when the sawing window will open. Previous research investigated the use of the ultrasonic pulse velocity (UPV) method to predict the saw-cutting window for early entry sawing. The results indicated that the method has the potential to provide effective guidance to contractors as to when to conduct early entry sawing. The aim of this project was to conduct similar work to observe the correlation between initial setting and conventional sawing time. Sixteen construction sites were visited in Minnesota and Missouri over a two-year period. At each site, initial set was determined using a p-wave propagation technique with a commercial device. Calorimetric data were collected using a commercial semi-adiabatic device at a majority of the sites. Concrete samples were collected in front of the paver and tested using both methods with equipment that was set up next to the pavement during paving. The data collected revealed that the UPV method looks promising for early entry and conventional sawing in the field, both early entry and conventional sawing times can be predicted for the range of mixtures tested.

Quantitative ultrasound imaging during shear wave propagation for application related to breast cancer diagnosis

Dans le contexte de la caractérisation des tissus mammaires, on peut se demander ce que l’examen d’un attribut en échographie quantitative (« quantitative ultrasound » - QUS) d’un milieu diffusant (tel un tissu biologique mou) pendant la propagation d’une onde de cisaillement ajoute à son pouvoir discriminant. Ce travail présente une étude du comportement variable temporel de trois paramètres statistiques (l’intensité moyenne, le paramètre de structure et le paramètre de regroupement des diffuseurs) d’un modèle général pour l’enveloppe écho de l’onde ultrasonore rétrodiffusée (c.-à-d., la K-distribution homodyne) sous la propagation des ondes de cisaillement. Des ondes de cisaillement transitoires ont été générés en utilisant la mèthode d’ imagerie de cisaillement supersonique ( «supersonic shear imaging » - SSI) dans trois fantômes in-vitro macroscopiquement homogènes imitant le sein avec des propriétés mécaniques différentes, et deux fantômes ex-vivo hétérogénes avec tumeurs de souris incluses dans un milieu environnant d’agargélatine. Une comparaison de l’étendue des trois paramètres de la K-distribution homodyne avec et sans propagation d’ondes de cisaillement a montré que les paramètres étaient significativement (p < 0,001) affectès par la propagation d’ondes de cisaillement dans les expériences in-vitro et ex-vivo. Les résultats ont également démontré que la plage dynamique des paramétres statistiques au cours de la propagation des ondes de cisaillement peut aider à discriminer (avec p < 0,001) les trois fantômes homogènes in-vitro les uns des autres, ainsi que les tumeurs de souris de leur milieu environnant dans les fantômes hétérogénes ex-vivo. De plus, un modéle de régression linéaire a été appliqué pour corréler la plage de l’intensité moyenne sous la propagation des ondes de cisaillement avec l’amplitude maximale de déplacement du « speckle » ultrasonore. La régression linéaire obtenue a été significative : fantômes in vitro : R2 = 0.98, p < 0,001 ; tumeurs ex-vivo : R2 = 0,56, p = 0,013 ; milieu environnant ex-vivo : R2 = 0,59, p = 0,009. En revanche, la régression linéaire n’a pas été aussi significative entre l’intensité moyenne sans propagation d’ondes de cisaillement et les propriétés mécaniques du milieu : fantômes in vitro : R2 = 0,07, p = 0,328, tumeurs ex-vivo : R2 = 0,55, p = 0,022 ; milieu environnant ex-vivo : R2 = 0,45, p = 0,047. Cette nouvelle approche peut fournir des informations supplémentaires à l’échographie quantitative statistique traditionnellement réalisée dans un cadre statique (c.-à-d., sans propagation d’ondes de cisaillement), par exemple, dans le contexte de l’imagerie ultrasonore en vue de la classification du cancer du sein.

Studies on some aspects of wave propagation through nonlinear media

Nonlinearity is a charming element of nature and Nonlinear Science has now become one of the most important tools for the fundamental understanding of the nature. Solitons— solutions of a class of nonlinear partial differential equations — which propagate without spreading and having particle— like properties represent one of the most striking aspects of nonlinear phenomena. The study of wave propagation through nonlinear media has wide applications in different branches of physics.Different mathematical techniques have been introduced to study nonlinear systems. The thesis deals with the study of some of the aspects of electromagnetic wave propagation through nonlinear media, viz, plasma and ferromagnets, using reductive perturbation method. The thesis contains 6 chapters

Rossby wave propagation on potential vorticity fronts with finite width

The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here an approximate correction is derived for the case of a finite width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.