Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Scaling laws for buckling of polar orthotropic annular plates subjected to compressive and torsional loading

Abstract: This paper derives the similitude invariants and scaling laws for buckling of polar orthotropic annular plates subjected to radial compressive load and torsional load. The similitude transformation is applied to the governing differential equation directly resulting in a scaling law for buckling load of annular plates and the similarity conditions between a model and a prototype. The scaling law is verified by a series of numerical tests using the solutions from the Ritz method as the theoretical solutions. For complete similitude cases, the buckling loads obtained from the scaling laws are identical to those of the theoretical solution. Since the derived scaling law is independent of the boundary conditions, it is valid for any pairs of model and prototype on the condition that they have identical boundary condition. Therefore, buckling behaviors of the prototype with complicated boundary conditions, of which a theoretical solution is not available, can be predicted from the experimental result on the model. Partial similitude model or scaling law for a pair of model and prototype without complete similarity is also investigated. The partial similarity model is not recommended for polar orthotropic materials, however it demonstrated good estimations for isotropic materials.

Viscous spreading of an inertial wave beam in a rotating fluid

Abstract: We report experimental measurements of inertial waves generated by an oscillating cylinder in a rotating fluid. The two-dimensional wave takes place in a stationary cross-shaped wavepacket. Velocity and vorticity fields in a vertical plane normal to the wavemaker are measured by a corotating particle image velocimetry system. The viscous spreading of the wave beam and the associated decay of the velocity and vorticity envelopes are characterized. They are found in good agreement with the similarity solution of a linear viscous theory, derived under a quasiparallel assumption similar to the classical analysis of Thomas and Stevenson [“A similarity solution for viscous internal waves,” J. Fluid Mech. 54, 495 (1972)] for internal waves.

Quasi-spherical, Time-dependent Viscous Accretion Flow: One-dimensional Results

Abstract: We investigated the instability of advective accretion flow as a consequence of angular momentum transfer in one-dimensional, quasi-spherical transonic accretion flow around a non-rotating black hole. The code is designed to include the effects of viscosity; the hydrodynamics component preserves angular momentum strictly with Lagrangian and remap method in the absence of viscosity, while the viscosity component updates viscous angular momentum transfer through the implicit method. We performed two tests to demonstrate the suitability of the code for accretion study. First, we simulated the inviscid, low angular momentum, transonic accretion flow with shocks around a black hole, and then the subsonic, self-similar ADAF solution around a Newtonian object. Both simulations fitted the corresponding analytical curves extremely well. We then simulated a rotating, viscous, transonic fluid with shocks. We showed that for low viscosity parameter, stable shocks at larger distance are possible. For higher viscosity parameter, more efficient angular momentum transfer in the post-shock disk makes the shock structure oscillatory. Moreover, as the shock drifts to larger distances, a secondary inner shock develops. We showed that the inner shock is the direct consequence of the expansion of the outer shock, as well as the creation of regions with ∂l/∂r < 0 due to more efficient angular momentum transfer near the inner sonic point. We showed that all disk parameters, including emissivity, oscillate with the same period as that of the shock oscillation. Our simulation may have implications for low frequency quasi-periodic oscillations, e.g., GRO J1655 – 40 and XTE J1550 – 564.