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Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings
... Therefore, approximate solutions such as alternate-line relaxation of the Jacobi or Gauss-Seidel relaxation can be used. In our implementation, we solve the residual equation using a dual time stepping technique (Jameson 1991). The method involves deriving a finite volume scheme by applying the relevant equations to control volumes to get a set of ordinary differential equations of the form, ...
... The coefficients α k and β k are chosen to maximize the stability interval along the imaginary and negative real axis of each Fourier mode of the solution. Jameson (1991) reported that effective choice for α and β are α 1 = 1/3, α 2 = 4/15, α 3 = 5/9, α 4 = 1 and β 1 = 1, β 2 = 1/2, β 3 = 0, and β 4 = 0. ...
Vortices, turbulence, and unsteady non-laminar flows are likely both prominent and dynamically important features of astrophysical disks. Such strongly nonlinear phenomena are often difficult, however, to simulate accurately, and are generally amenable to analytic treatment only in idealized form. In this paper, we explore the evolution of compressible two-dimensional flows using an implicit dual-time hydrodynamical scheme that strictly conserves vorticity (if applied to simulate inviscid flows for which Kelvin's Circulation Theorem is applicable). The algorithm is based on the work of Lerat, Falissard & Side (2007), who proposed it in the context of terrestrial applications such as the blade-vortex interactions generated by helicopter rotors. We present several tests of Lerat et al.'s vorticity-preserving approach, which we have implemented to second-order accuracy, providing side-by-side comparisons with other algorithms that are frequently used in protostellar disk simulations. The comparison codes include one based on explicit, second-order van-Leer advection, one based on spectral methods, and another that implements a higher-order Godunov solver. Our results suggest that Lerat et al's algorithm will be useful for simulations of astrophysical environments in which vortices play a dynamical role, and where strong shocks are not expected.
... More detailed information can be found in the literature [20]. 3 × 10 6 4 In addition, given that flow steadiness is the main focus of this study, the temporal discretization uses the canonical LU-SGS method with a local time step to efficiently determine flow steadiness [24]. ...
The evolution and sudden change of steady interaction structures is numerically studied with the fore wedge angle theta_1 in a low enthalpy hypersonic double wedge configuration. It particularly focuses on the conditions of Swantek and Austin's experiments where Ma=7, and h_0=2 MJ/kg but with a reduced Reynolds number (Re). The sudden structural change indicates that when theta_1 reaches a critical value, minor angular variations can trigger a discontinuous transformation in flow structures. The analysis is based on the laminar Navier-Stokes equations, using ideal gas and non-equilibrium gas models. Under the condition of Re=1E5/m, detailed numerical simulations are conducted as theta_1 varies over 0 deg-40 deg. This study yields the following findings: (a) The upper and lower boundaries of theta_1 for the onset of unsteady flow are identified. When theta_1 lies outside these boundaries, the flow remains steady. (b) As theta_1 increases, the interaction patterns evolve sequentially, progressing from Type VI through Type VI->V, Type III, Type IV_r, and ultimately to a flow dominated solely by a bow shock. This evolution defines the boundaries between different interaction patterns and provides a comprehensive understanding of their progression with theta_1. Sudden structural changes occur during the transitions from Type III to Type IV_r and from Type IV_r to a bow shock-dominated flow. In addition, a comparative study is performed through shock polar analysis to compare its predictions with computational results. (c) An unconventional reflection pattern of the transmitted shock over the separation zone, called Type III_r, is observed in non-equilibrium gas flows, which differs from classical interaction patterns. (d) The aerodynamic distribution of wall properties under various interactions is obtained, indicating distinct features before and after the sudden structural change.
... Roe's approximate Riemann solver and the Monotone Upwind Scheme for Conservation Law extrapolation are implemented to calculate convective terms and central differences for the diffusion fluxes (Roe, 1981). The unsteady flow computations are conducted using Jameson's dual time-stepping method for an implicit scheme with 15 sub-iterations (Jameson, 1991). The wellproven Menter Shear Stress Transport (SST) Turbulence Model is used for the calculation. ...
The circumferential non-uniform tip clearance (CNTC) due to casing out-of-roundness adversely affects the turbine aerodynamic performance due to machining and assembly errors, thermal deformation, and improper active clearance control (ACC), etc. Moreover, the asymmetric computational domain caused by casing out-of-roundness presents difficulties for conventional numerical techniques that consider rotational periodicity. Since previous traditional methods using split computational domains have the disadvantages of high interpolation error and high time cost, an efficient fast-moving mesh (FMM) method based on an algebraic approach is proposed in this paper. This method is first validated by using a single-stage turbine with elliptical casing. The results show that the FMM has the advantages of high accuracy, high efficiency, and easy operation, which helps to solve the CNTC problem quickly in scientific research or engineering applications. Then, the effects of CNTC induced by the elliptical casing on the flow field and aerodynamic performance are investigated by using an in-house code that integrates the FMM method. Finally, the effect of stator row interference on the aerodynamic performance in the turbine stage with an elliptical casing is demonstrated. The results show that different types of elliptical casings have a significant effect on the aerodynamic performance. However, the variation law is not consistent (decreasing by 0.538% or increasing by 0.212%). Importantly, the novel finding of this paper is that this discrepancy is jointly determined by the interaction of multiple secondary flows (passage vortex, scraping vortex, etc.) at different spans, not just related to the variation of the tip leakage vortex (TLV) with tip size. Furthermore, this study is the first to indicate that the stator row interference can mitigate the extent of performance degradation due to elliptical casings by suppressing the development of secondary flows. These results may provide theoretical support for blade tip gap design and can also serve as a reasonable reference for the effective application of ACC in engineering. Finally, low-order harmonic components with high amplitudes are also innovatively found in the rotor row with a CNTC. These components may cause low-engine-order (LEO) resonances that endanger the safe operation of engines. Article History
... , as well as certain design parameters ρ ∈ R p that determine the PDE state. As implicit time integration methods are often preferred due to better stability properties, one application of the time-stepper Φ i will likely require solving nonlinear and linear equations iteratively at each time step, as for example in a dual time-stepping framework [25]. In order to facilitate readability, the dependency on previous time steps other than u i−1 will be dropped for the rest of this paper. ...
This paper presents a non-intrusive framework for integrating existing unsteady partial differential equation (PDE) solvers into a parallel-in-time simultaneous optimization algorithm. The time-parallelization is provided by the non-intrusive software library XBraid, which applies an iterative multigrid reduction technique to the time domain of existing time-marching schemes for solving unsteady PDEs. Its general user-interface has been extended for computing adjoint sensitivities such that gradients of output quantities with respect to design changes can be computed parallel-in-time alongside with the primal PDE solution. In this paper, the primal and adjoint XBraid iterations are embedded into a simultaneous optimization framework, namely the One-shot method. In this method, design updates towards optimality are employed after each state and adjoint update such that optimality and feasibility of the design and the PDE solution are reached simultaneously. The time-parallel optimization method is validated on an advection-dominated flow control problem which shows significant speedup over a classical time-serial optimization algorithm.
... Such multigrid processing could turn out to be very important in designing time-implicit schemes for MHD. The work of Jameson [28] and Gropp et al [25] has already demonstrated the importance of multilevel processing for the Euler equations. The work of Balsara [6] has already demonstrated the importance of multigrid methods for the equations of radiative transfer. ...
In this paper we present a full-fledged scheme for the second order accurate, divergence-free evolution of vector fields on an adaptive mesh refinement (AMR) hierarchy. We focus here on adaptive mesh MHD. The scheme is based on making a significant advance in the divergence-free reconstruction of vector fields. In that sense, it complements the earlier work of Balsara and Spicer (1999) where we discussed the divergence-free time-update of vector fields which satisfy Stoke's law type evolution equations. Our advance in divergence-free reconstruction of vector fields is such that it reduces to the total variation diminishing (TVD) property for one-dimensional evolution and yet goes beyond it in multiple dimensions. Divergence-free restriction is also discussed. An electric field correction strategy is presented for use on AMR meshes. The electric field correction strategy helps preserve the divergence-free evolution of the magnetic field even when the time steps are sub-cycled on refined meshes. The above-mentioned innovations have been implemented in Balsara's RIEMANN framework for parallel, self-adaptive computational astrophysics which supports both non-relativistic and relativistic MHD. Several rigorous, three dimensional AMR-MHD test problems with strong discontinuities have been run with the RIEMANN framework showing that the strategy works very well.
... The time integration is carried out by an explicit hybrid 5-stage Runge-Kutta scheme. Dual time-stepping according to Jameson [23], multigrid level 3 and implicit residual smoothing with variable coefficients were applied. The CFL numbers are 6.5 and 1.5 for the fine and coarse grid levels, respectively, in the multigrid scheme. ...
Computational fluid dynamics (CFD) studies are carried out on a two-bladed vertical axis wind turbine (VAWT) operating at a wind speed of 8 m/s for the tip speed ratios () of 0.50 - 3.0. The blade consists of the NACA 0021 airfoil with the chord length of 0.265 m and a rotor radius of 1 m. Basic sensitivity studies for various time step sizes are carried out. The results are validated against available measurement data from literature. An excellent agreement is obtained for small up to optimum condition. For the higher tip speed ratios, the two dimensional CFD computations predict higher results than the wind tunnel experiment, but they are very similar to the field measurement data. Wake characteristics are presented in the present studies, showing that the wake becomes Gaussian at 5 times radius downsteam of the rotor. It is shown that complex flow phenomena occur due to dynamic stall onset especially for the smaller tip speed ratio.
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