Effects of the Jacobian evaluation on Newton's solution of the Euler equations

SUMMARYThis paper makes the first attempt of extending implicit AUSM‐family schemes to multiphase flow simulations. Water faucet, air–water shock tube and oscillating manometer problems are used as benchmark tests with the generic four‐equation two‐fluid model. For solving the equations implicitly, Newton's method along with a sparse matrix solver (UMFPACK solver) is employed, and the numerical Jacobian matrix is calculated. Comparison between implicit and explicit AUSM‐family schemes is presented, indicating that similarly accurate results are obtained with both schemes. Furthermore, the water faucet problem is solved using both staggered and collocated grids. This investigation helps integrate high‐resolution schemes into staggered‐grid‐based computational algorithms. The influence of the interface pressure correction on the simulation results is also examined. Results show that the interfacial pressure correction introduces numerical dissipation. However, this dissipation cannot eliminate the overshoots because of the incompatibility of numerical discretization of the conservative and non‐conservative terms in the governing equations. The comparison of CPU time between implicit and explicit schemes is also studied, indicating that the implicit scheme is capable of improving the computational efficiency over its explicit counterpart. Copyright © 2014 John Wiley & Sons, Ltd.

“…Here, the numerical Jacobian calculation will be used for the derivative term in Equation :

{((), \frac{\partial {boldR}_{j}}{\partial {boldw}_{i}})}^{m} = \frac{{boldR}_{j} ({boldw}^{m} + ϵ_{i} {bolde}_{i}) - {boldR}_{j} ({boldw}^{m})}{ϵ_{i}}

Section: Methodsmentioning

confidence: 99%

“…Here, the numerical Jacobian calculation will be used for the derivative term in Equation (45) [21]:…”

Section: Numerical Jacobian Calculationmentioning

confidence: 99%

Comparison of implicit and explicit AUSM‐family schemes for compressible multiphase flows

Zeng

Aydemir

Lien

et al. 2014

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

Gelfgat

2006

SUMMARYA problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second-order finite volume method. The study is motivated by further applications of the finite volumebased stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low-order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov-subspace-based iteration techniques. The same replacement in the Newton steady-state solver also yields a robust numerical process, however, it cannot be as effective as modern preconditioned Krylov-subspace-based iterative solvers.

Numerical continuation of high Reynolds number external flows

Wales¹,

Gaitonde²,

Jones³

et al. 2010

SUMMARYA numerical continuation method for the compressible Reynolds-Averaged Navier-Stokes equation with the Spalart-Allmaras turbulence model is presented and applied to the flow around a 2D airfoil. Using continuation methods it is possible to study the steady flow states of a system as a parameter such as angle of attack is varied. This approach allows unstable solutions to be calculated, which are important for understanding the nonlinear dynamics of the system. Furthermore, this method can be used to find any multivalued solutions that exist at a single parameter value. The eigenvalues of the system are calculated using the Cayley transform to precondition the eigenvalue solver ARPACK. The eigenvalues are important as they show the stability of the solutions as well as accurately detect parameter values at which bifurcations take place.