Transient growth of perturbations by a linear non-modal evolution is studied here in a stably stratified bounded Couette flow. The density stratification is linear. Classical inviscid stability theory states that a parallel shear flow is stable to exponentially growing disturbances if the Richardson number (Ri) is greater than 1/4 everywhere in the flow. Experiments and numerical simulations at higher Ri show however that algebraically growing disturbances can lead to transient amplification. The complexity of a stably stratified shear flow stems from its ability to combine this transient amplification with propagating internal gravity waves (IGWs). The optimal perturbations associated with maximum energy amplification are numerically obtained at intermediate Reynolds numbers. It is shown that in this wall-bounded flow, the threedimensional optimal perturbations are oblique, unlike in unstratified flow. A partitioning of energy into kinetic and potential helps in understanding the exchange of energies and how it modifies the transient growth. We show that the apportionment between potential and kinetic energy depends, in an interesting manner, on the Richardson number, and on time, as the transient growth proceeds from an optimal perturbation. The oft-quoted stabilizing role of stratification is also probed in the non-diffusive limit in the context of disturbance energy amplification.
Introduction: Myriad disorders cause right ventricular (RV) dilation and lead to tricuspid regurgitation (TR). Because the thin-walled, flexible RV is mechanically coupled to the pulmonary circulation and the left ventricular septum, it distorts with any disturbance in the cardiopulmonary system. TR, therefore, can result from pulmonary hypertension, left heart failure, or intrinsic RV dysfunction; but once it occurs, TR initiates a cycle of worsening RV volume overload, potentially progressing to right heart failure. Characteristic three-dimensional RV shape-changes from this process, and changes particular to individual TR causes, have not been defined in detail.Methods: Cardiac MRI was obtained in 6 healthy volunteers, 41 patients with ≥ moderate TR, and 31 control patients with cardiac disease without TR. The mean shape of each group was constructed using a three-dimensional statistical shape model via the particle-based shape modeling approach. Changes in shape were examined across pulmonary hypertension and congestive heart failure subgroups using principal component analysis (PCA). A logistic regression approach based on these PCA modes identified patients with TR using RV shape alone.Results: Mean RV shape in patients with TR exhibited free wall bulging, narrowing of the base, and blunting of the RV apex compared to controls (p<0.05). Using four primary PCA modes, a logistic regression algorithm identified patients with TR correctly with 82% recall and 87% precision. In patients with pulmonary hypertension without TR, RV shape was narrower and more streamlined than in healthy volunteers. However, in RVs with TR and pulmonary hypertension, overall RV shape continued to demonstrate the free wall bulging characteristic of TR. In the subgroup of patients with congestive heart failure without TR, this intermediate state of RV muscular hypertrophy was not present.Conclusion: The multiple causes of TR examined in this study changed RV shape in similar ways. Logistic regression classification based on these shape changes reliably identified patients with TR regardless of etiology. Furthermore, pulmonary hypertension without TR had unique shape features, described here as the “well compensated” RV. These results suggest shape modeling as a promising tool for defining severity of RV disease and risk of decompensation, particularly in patients with pulmonary hypertension.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.