High-Resolution and High-Order CFD Methods

1. High-Resolution Methods for Incompressible Flows

We have developed an algorithm that unifies all the well known incompressible flow methods [1]. The algorithm combines the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations, thus enabling the solution to take advantage of best numerical features obtained from the above methods. The algorithm (labelled as FSAC-PP) also employs high-resolution and high-order methods [2], e.g. the characteristics-based (CB) Godunov-type treatment of convective terms by Drikakis et al. [3-7]. FSAC-PP unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers; b) exploit the excellent accuracy and convergence properties of the CB schemes in the framework of PP methods; and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PP method has also been coupled with a non-linear, full-multigrid and full-approximation storage (FMG-FAS) technique [7] to further increase the efficiency of the solution. Validation of the FSAC-PP method has been achieved for several computational examples [1]. The results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

  • L. Konozsy, D. Drikakis, A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations, Communications in Computational Physics, 2014, accepted (in print).
  • D. Drikakis and W. Rider High-Resolution Methods for Incompressible and Low-Speed Flows, Springer, 2005, 622 pages CFD textbook, (ISBN: 3-540-22136-0).
  • E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit, Journal of Computational Physics, 210, 584-607, 2005.
  • E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part II. Multigrid implementation and numerical tests, Journal of Computational Physics, 210, 608-631, 2005.
  • D. Drikakis, P. Govatsos, D. Papantonis, A Characteristic Based Method for Incompressible Flows, International Journal for Numerical Methods in Fluids, Vol. 19, 667-685, 1994.
  • E. Shapiro, D. Drikakis, Non-conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows, International Journal for Numerical Methods in Engineering, Vol. 66, 9, 1466-1482, 2006.
  • D. Drikakis, O. Iliev, D.P. Vassileva, A non-linear full multigrid method for the three-dimensional incompressible Navier-Stokes equations, Journal of Computational Physics, 146, 301-321, 1998.
High-resolution simulation of instability and transition to turbulence behind a stenosis

2. Improved methods for compressible flows with low Mach features

We have developed a simple modification of the variable reconstruction process within finite volume schemes to allow significantly improved resolution of low Mach number perturbations for use in mixed compressible/incompressible flows. The main advantage is that the numerical method locally adapts the variable reconstruction to allow minimum dissipation of low Mach number features whilst maintaining shock capturing ability, all without modifying the formulation of the governing equations. In addition, incompressible scaling of the pressure and density variations are recovered. Numerical tests using a Godunov-type method demonstrate that the new scheme captures shock waves well, significantly improves resolution of low Mach number features and greatly reduces high wave number dissipation in the case of homogeneous decaying turbulence and Richtmyer-Meshkov mixing. In the latter case, the turbulent spectra match theoretical predictions excellently. Additional computational expense due to the proposed modification is negligible.

Richtmyer-Meshkov turbulent mixing without (left) and with (right) low Mach corrections . The numerical spectra obtained from simulations using a fifth-order high-resolution scheme are also compared with the theoretical predictions.


  • B. Thornber, A. Mosedale, D. Drikakis, D. Youngs, R. Williams, An Improved Reconstruction Method for Compressible Flows with Low Mach Number Features, Journal of Computational Physics, 227, 4873-4894, 2008
  • B. Thornber, D. Drikakis, R. Williams, D. Youngs, On Entropy Generation and Dissipation of Kinetic Energy in High-Resolution Shock-Capturing Schemes, Journal of Computational Physics, 227, 4853-4872, 2008.
  • D. Drikakis, M. Hahn, A. Mosedale, B. Thornber, Large Eddy Simulation Using High Resolution and High Order Methods, Philosophical Transactions Royal Society A, 367, 2985-2997, 2009.
  • A. Mosedale, D. Drikakis, Assessment of very high-order of accuracy in LES models, ASME Journal of Fluids Engineering, Vol. 129(12), 1497-1503, 2007.

3. High-order Methods for laminar, transitional and turbulent compressible flows on unstructured and hybrid meshes

The work concerns the development of weighted-essentially-non-oscillatory (WENO) schemes for viscous compressible flows on arbitrary unstructured grids. WENO schemes up to fifth-order accurate have been developed in conjunction with hybrid and fully unstructured grids. The schemes have been validated with reference to numerical and experimental results for a range of problems ranging from canonical flows, such as the the TaylorGreen vortex, as well as for laminar and turbulent subsonic flows, and turbulent shock-wave boundary layer interaction. The research has shown that the accuracy of the schemes depends on the arbitrariness of shape and orientation of the unstructured mesh elements, as well as the compactness of directional stencils. The WENO schemes provide a more accurate numerical framework compared to second-order and third-order total variation diminishing (TVD) methods. The third-order variant offers an excellent numerical framework in terms of accuracy and computational cost compared to the fifth-order WENO and second-order TVD schemes. Parallelisation of the CFD code (UCNS3D), where the schemes have been implemented, shows that the present methods offer very good scalable performance.

Top: Turbulent flow around a sphere at Re=10,000. Bottom: Supersonic shock-wave/turbulent boundary layer interaction.


  • P. Tsoutsanis, A.F. Antoniadis, D. Drikakis, WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flow, Journal of Computational Physics, 256, 254-276, 2014.
  • P. Tsoutsanis, V.A. Titarev, D. Drikakis. WENO schemes on arbitrary mixed element unstructured meshes in three space dimensions, Journal of Computational Physics, Vol. 230, 4(20), 1585-1601, 2011.