A Diagonally Implicit Runge-Kutta Method for the Discontinuous Galerkin solutions of the Navier-Stokes Equations

2011 ◽  
Author(s):  
Hidehiro Segawa ◽  
Hong Luo ◽  
Robert Nourgaliev
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Kutta Method ◽  
Navier Stokes Equations ◽  
Runge Kutta Method ◽  
Runge Kutta

Calculation of the Maximal Flight Altitude of the Historical HW2 Rocket Using CFD

2018 ◽  
Author(s):  
Michael Steppert ◽  
Philipp Epple ◽  
Michael Steber
Keyword(s):  
Stokes Equations ◽  
Rocket Engine ◽  
Navier Stokes ◽  
Kutta Method ◽  
Flight Altitude ◽  
Atmospheric Conditions ◽  
Flight Trajectory ◽  
Navier Stokes Equations ◽  
Technical Problems ◽  
Runge Kutta Method

The historical HW2 rocket was a liquid propulsion rocket, designed by the German rocket pioneer Johannes Winkler in 1932. With this rocket, Winkler tried to reach a much higher altitude than with his first model, the HW1, which was the first liquid propulsion rocket in Europe and reached an altitude of 60 meters. Because of technical problems, the HW2 exploded immediately after the launch on October 6th in 1932 [1] [2]. To estimate the performance of this historical liquid propulsion rocket its maximum flight altitude was computed with the use of CFD. The equation of the vertical flight trajectory was solved numerically, with the classical Runge-Kutta method. For the computation of the vertical trajectory standard atmospheric conditions were considered. To determine the thrust and the drag of the rocket, the Navier-Stokes equations were solved with the commercial CFD solver Star-CCM+ from Siemens PLM Software. The rocket hull and the rocket engine were first simulated independently for different Mach-numbers and atmospheric flight conditions. Finally the complete rocket with running rocket engine was also computed in atmospheric flight conditions. These results were compared with the standalone simulations of the rocket drag without the running rocket engine and with the simulation of the rocket engine alone. The results are shown and analyzed in detail in this work.

scholarly journals A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

2016 ◽  
Vol 20 (4) ◽  
pp. 1016-1044 ◽  
Author(s):  
Xiaodong Liu ◽  
Yidong Xia ◽  
Hong Luo ◽  
Lijun Xuan
Keyword(s):  
Comparative Study ◽  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Index 2 ◽  
Runge Kutta

AbstractA comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part I: Implicit Runge–Kutta Schemes

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Graham Ashcroft ◽  
Christian Frey ◽  
Kathrin Heitkamp ◽  
Christian Weckmüller
Keyword(s):  
Stokes Equations ◽  
Temporal Integration ◽  
Aerodynamic Noise ◽  
Navier Stokes ◽  
Noise Generation ◽  
Academic Problems ◽  
Tonal Noise ◽  
Navier Stokes Equations ◽  
Runge Kutta ◽  
The Stability

This is the first part of a series of two papers on unsteady computational fluid dynamics (CFD) methods for the numerical simulation of aerodynamic noise generation and propagation. In this part, the stability, accuracy, and efficiency of implicit Runge–Kutta schemes for the temporal integration of the compressible Navier–Stokes equations are investigated in the context of a CFD code for turbomachinery applications. Using two model academic problems, the properties of two explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) schemes of second- and third-order accuracy are quantified and compared with more conventional second-order multistep methods. Finally, to assess the ESDIRK schemes in the context of an industrially relevant configuration, the schemes are applied to predict the tonal noise generation and transmission in a modern high bypass ratio fan stage and comparisons with the corresponding experimental data are provided.

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