Biomechanical and functional alterations in the diabetic foot: Differences between type I and type II Diabetes

Abstract

A shock-accelerated Refrigerant-22 bubble is computed using third-, fifth- and ninth-order accurate upwind schemes and a fifth-order accurate Weighted Essentially Non-Oscillatory (WENO) scheme by solving the Navier–Stokes equations. A third-order accurate four-stage Runge–Kutta scheme is used for time integration. The Mach 1.22 shock excites Richtmyer–Meshkov instability at the bubble-air interface, and the resulting small-scale vortices are resolved well by the high-resolution schemes due to their low numerical dissipation. Initial wave patterns and bubble shape match with existing experimental and numerical results. Differences become visible between the results of the different schemes at later times when the interfacial vortices grow into localized turbulent mixing zones. A central jet appears after the departure of the refracted shock from the bubble as a transmitted shock. The appearance of this jet confirms that lack of gas viscosity in earlier Euler simulations was not the reason behind its magnified appearance compared to experiments. The computations have been carried out upto 1539.22 µs after the shock first touches the upstream bubble-air interface. Results for such long time duration are not available in existing literature, to the best of our knowledge. Earlier simulation of this problem was carried out by solving the Euler equations. By solving the Navier–Stokes equations, we do not ignore gas viscosity, the effect of which might be significant at the late stages of the evolving bubble. The fifth-order WENO consistently predicts less mixing compared to the fifth-order upwind scheme. Enstrophy predicted by different schemes start varying significantly from each other after the refracted shock emerges from the downstream bubble-air interface. Difference in enstrophy between the ninth- and third-order schemes is found to be as high as 30% near the end of the simulation time (about 1539 µs). Higher-order schemes show more small-scale vortices embedded within the large-scale structures, and between the two fifth-order schemes, the WENO shows less proliferation of small eddies. The presented results indicate that the ninth-order scheme can be a fairly good choice for long time computation of shock-deformed R22 bubbles, and that the standard Jiang-Shu fifth WENO scheme can visibly and significantly deviate from the plain fifth-order upwind scheme in such long time integration problems.