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Numerical Methods for Inviscid Flow Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
Shock-capturing schemes are the most widely used techniques for computing inviscid flows with shocks. In this approach, the Euler equations are cast in conservation-law form, and any shock waves or other discontinuities are computed as part of the solution. The shock waves predicted by these methods are usually smeared over several mesh intervals, but the simplicity of the approach may outweigh the slight compromise in results compared to the more elaborate shock-fitting schemes. Classical shock-capturing methods have the disadvantage that very strong shocks will cause the methods to fail. This failure is usually evidenced by oscillations. Computations in hypersonic flow with very strong shocks typically lead to the appearance of negative pressures and subsequent divergence of the solution during the time-dependent computation process. In addition to this problem, higher-order schemes tend to produce oscillations in the solution. However, these methods are useful and will be modified in later sections to avoid these difficulties. The alternative approach is to fit each shock wave as a discontinuity and solve for the discontinuity as part of the solution. This shock-fitting approach is very elegant and produces shocks that are truly discontinuous. Unfortunately, the procedure for general shock fitting in three dimensions with multiple shocks is extremely complex, and as a result, the use of shock fitting is usually limited to fitting shocks at boundaries.
Implicit and explicit TVD methods for discontinuous open channel flows
Published in R. A. Falconer, K. Shiono, R. G. S. Matthew, Hydraulic and Environmental Modelling: Estuarine and River Waters, 2019
The use of numerical methods to predict the water profile and discharge for unsteady as well as for stationary situations of hydraulic systems is now of common use in engineering work. Most difficult situations occur when mixed sub-supercritical regimes with hydraulic jumps are present in the system, that invalidate some of the numerical methods available for calculation. Solutions of that kind usually appear when modelling steady flows in steeply sloping channels, rapidly varied steady and unsteady flows, or in dam collapse simulations. Among the techniques that succeed in those difficult cases are the through or shock capturing methods in which the equations governing the model are solved in conservation form by means of a suitable numerical scheme. This approach can locate and propagate the discontinuities present in the solution with the physically correct speed and strength without the need for any a priori information or fitting procedure.
The elusive topic of rainfall-induced surge waves in rivers: lessons from canyon accidents
Published in International Journal of River Basin Management, 2022
The numerical integration here performed is particularly meaningful for the present purposes because: it is characterized by relevant numerical diffusion effects, that would tend to counteract the steepening tendency and the development of the shock;the shock-capturing methods can simulate steep front waves and shocks without introducing further conditions or procedures. Previous calculations related to hydrograph steepening (Collischonn et al., 2017; Ponce & Windingland, 1985) are not based on shock-capturing methods, that have rather been used for simulating surge wave propagation in sewers (e.g. Lautenbach et al., 2008);the complete form of the unsteady flow equations here adopted ensure that the effects of approximations (including kinematic hypothesis) are ruled out.
Computational modelling of multi-material energetic materials and systems
Published in Combustion Theory and Modelling, 2020
Alberto M. Hernández, D. Scott Stewart
There are two main approaches for solving hyperbolic conservation laws: shock capturing and front tracking methods. For Conservation laws (i.e. Euler Equations in conservation form) shock capturing schemes inherently capture discontinuities, unlike shock tracking methods that need to use the Rankine-Hugoniot relations. But due to numerical truncation errors, shock capturing methods introduce diffusion across discontinuities. These can cause issues in flow models where sharp changes in material properties (large gradients) as well as different constitutive relationships in the form of equations of states are present. Cocchi [1,33] pointed out that when numerical diffusion is present, density diffuses across the interface requiring a mixture equation of state to describe and maintain a continuous pressure field in this non-physical region. For the multi-component reactive flow models of interest in this paper, having a mixture equation of state leads to similar convergence issues as explained in detail in Section 2.1.2. Spurious pressure oscillations are also observed when applying these types of schemes. These drawbacks were part of the motivation behind the development of higher order schemes.
Dynamic mode decomposition of supersonic turbulent pipe flow with and without shock train
Published in Journal of Turbulence, 2020
G. Srinivasan, Susila Mahapatra, Kalyan P. Sinhamahapatra, Somnath Ghosh
However, there are very few LES studies of shock–turbulence interaction in internal flows. Koo and Raman [14] studied shock–turbulence interaction in a supersonic inlet isolator using LES. Different shock capturing methods like hyperviscosity approach and weighted essentially non-oscillatory (WENO) schemes were used and compared. LES of shock train in a constant area supersonic isolator with rectangular cross-section has been performed by Morgan and Lele [15] and the results were compared qualitatively with experiments carried out at a higher Reynolds number. Using the ALDM approach, Quaatz et al. [16] studied shock train in a Laval nozzle with rectangular cross-section. Mahapatra et al. [17] studied shock train phenomena in supersonic pipe and divergent–convergent diffuser flows with isothermal walls by means of LES. The shock train showed low-frequency oscillations which was demonstrated and quantified in the study by means of pressure traces and power-spectral density.