Extension of Harten-Lax-van Leer Scheme for Flows at All Speeds.
The Harten, Lax, and van Leer with contact restoration (HLLC) scheme has been modified and extended in conjunction with time-derivative preconditioning to compute flow problems at all speeds. It is found that a simple modification of signal velocities in the HLLC scheme based on the eigenvalues of the preconditioned system is only needed to reduce excessive numerical diffusion at the low Mach number. The modified scheme has been implemented and used to compute a variety of flow problems in both two and three dimensions on unstructured grids. Numerical results obtained indicate that the modified HLLC scheme is accurate, robust, and efficient for flow calculations across the Mach-number range. ISTORICALLY, numerical algorithms for the solution of the Euler and Navier‐Stokes equations are classified as either pressure-based or density-based solution methods. The pressurebased methods, originally developed and well suited for incompressible flows, are typically based on the pressure correction techniques. They usually use a staggered grid and solve the governing equations in a segregated manner. The density-based methods, originally developed and robust for compressible flows, use time-arching procedures to solve the hyperbolic system of governing equations in a coupled manner. In general, density-based methods are not suitable for efficiently solving low Mach number or incompressible flow problems, because of large ratio of acoustic and convective timescales at the low-speed flow regimes. To alleviate this stiffness and associated convergence problems, time-derivative preconditioning techniques have been developed and used successfully for solving low-Machnumber and incompressible flows by many investigators, including Chorin, 1 Choi and Merkle, 2 Turkel, 3 Weiss and Smith, 4 and Dailey and Pletcher, 5 among others. Such methods seek to modify the time component of the governing equations so that the convergence can be made independent of Mach number. This is accomplished by altering the acoustic speeds of the system so that all eigenvalues become of the same order, and thus condition number remains bounded independent of the Mach number of the flows. Over the last two decades characteristic-based upwind methods have established themselves as the methods of choice for prescribing the numerical fluxes for compressible Euler equations. When these upwind methods are used to compute the numerical fluxes for the preconditioned Euler equations, solution accuracy at low speeds can be compromised, unless the numerical flux formulation is modified to take into account the eigensystem of the precondi