After casting steel slabs are reheated in a reheat furnace to temperatures in the range 1200-1250°C in order to be suitable for rolling. The high energy requirements and the importance of reheating for quality control are the motivation behind numerically modelling the furnace. Computational fluid dynamics allows us to understand the fundamental physics with great detail. It is however unclear how assumptions of such models influence the results of the simulations. In this work a steady-state model was analysed and it was found that the chosen slab temperature profile can underestimate the average heat flux on the slab surface by 30%. A slab model was employed to simulate the transient slab temperatures which results in an underestimation of the average slab temperatures by about 500°C for the case with reduced fluxes. The uniform slab temperature assumption also results in the overestimation of heat fluxes on its front and side face.
Steady free surface flows are of interest in the fields of marine and hydraulic engineering. Fitting methods are generally used to represent the free surface position with a deforming grid. Existing fitting methods tend to use time-stepping schemes, which is inefficient for steady flows. There also exists a steady iterative method, but that one needs to be implemented with a dedicated solver.Therefore a new method is proposed to efficiently simulate two-dimensional (2D) steady free surface flows, suitable for use in conjunction with black-box flow solvers. The free surface position is calculated with a quasi-Newton method, where the approximate Jacobian is constructed in a novel way by combining data from past iterations with an analytical model based on a perturbation analysis of a potential flow. The method is tested on two 2D cases: the flow over a bottom topography and the flow over a hydrofoil. For all simulations the new method converges exponentially and in few iterations. Furthermore, convergence is independent of the free surface mesh size for all tests. K E Y W O R D Sfitting method, free surface flow, perturbation analysis, quasi-Newton INTRODUCTIONThis paper considers the numerical simulation of steady free surface flow of incompressible, immiscible fluids with a large density difference, typically water and air. This type of flow is often encountered in the fields of marine and hydraulic engineering, for example to calculate ship hull resistance, analyze ship-wall interactions in narrow straight canals and study flow behavior in river confluences. These flows are governed by the incompressible Navier-Stokes equations, which are typically solved with computational fluid dynamics (CFD). The free surface causes additional difficulties for the flow computations: its position is unknown a priori, and needs to be determined as a component of the solution during the computation. Two classes of methods exist to represent the free surface, namely, surface capturing and surface fitting methods. * In surface capturing approaches, the computational mesh is not aligned with the interface, and the interface can intersect the mesh in an in principle arbitrary manner. Capturing methods are versatile in that they can handle complex phenomena such as wave breaking. Examples are the marker-and-cell method, 2 the volume-of-fluid method, 3 and the level-set method. 4 *Classification and terminology of these methods tends to be inconsistent in the literature; terminology is adopted from Wackers et al. 1Int J Numer Meth Fluids. 2020;92:785-801.wileyonlinelibrary.com/journal/fld
Fluid-structure interaction (FSI) problems are frequently solved using partitioned simulation techniques with black-box solvers, reusing reliable and optimized codes. These problems can principally be reduced to solving a root-finding problem. In case of strong coupling, pure Gauss-Seidel iterations between the structure and flow solvers are unstable for lower modes. In these cases, quasi-Newton techniques are used, which construct an approximation of the Jacobian or its inverse by reusing information from previous iterations and time steps. Four different quasi-Newton techniques are compared: the interface quasi-Newton algorithm with an approximation for the inverse of the Jacobian from a least-squares model (IQN-ILS), the interface block quasi-Newton algorithm with approximate Jacobians from least-squares models (IBQN-LS), the interface quasi-Newton technique with multiple vector Jacobian (IQN-MVJ) and the multi-vector update quasi-Newton technique (MVQN). These coupling algorithms are differentiated based on whether the approximation of the Jacobian is performed for the entire black-box system (IQN-ILS and IQN-MVJ) or for both individual solvers (IBQN-LS and MVQN). Moreover, a distinction is made between methods which perform the approximation with either least-squares models (IQN-ILS and IBQN-LS) or multivector techniques (IQN-MVJ and MVQN). Their performance is compared by solving a 1D flexible tube case, using the in-house coupling software CoCoNuT. Both the memory usage and number of iterations between structure and flow solvers in each time step are examined. The techniques using a multi-vector approach require explicit matrix construction, so that memory requirements scale quadratically, whereas the least-squares techniques have a matrix-free implementation, resulting in linear scaling. In terms of convergence they are comparable.
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