Dissipation of upwind schemes at high wave numbers

geraldine.fjfi.cvut.cz

The work deals with numerical solution of two compressible flows problems. Firstly authors considered steady transonic flows through DCA 8% cascade (Double Circular Arc symmetrical) for increasing upstream Mach numbers M ∞ ∈ (0.813; 1.13). The cascade flows were suggested in Institute of Thermomechanics by Mr. Dvořák and flows were investigated experimentally. The structure of flow seems to be very complicated. It is possible to observe subsonic and supersonic part, shock wave structure, interaction of shock wave and boundary layer, wake etc. We investigated these flows numerically using composite scheme in the form of finite volume method for governing system of Euler equations. These numerical results are compared to experimental data of IT CAS CZ using comparison of several regimes with increasing upstream Mach numbers. The second problem is an unsteady viscous flow with very low upstream Mach number M∞ ≈ 0.02 in a 2D channel with a moving part of solid wall as a function of time. The flow is described by the system of Navier-Stokes equations for compressible laminar flows. The problem is numerically solved by MacCormack finite volume scheme. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method.

Computers & fluids, 1998

a b s t r a c t High-accuracy, time-accurate compressible Navier-Stokes solvers have been developed for transonic flows. These solvers use optimized upwind compact schemes (OUCS) and four-stage, fourth order explicit Runge-Kutta (RK4) time integration scheme, details of which can be obtained in Sengupta [Sengupta TK. High Accuracy Computing Methods, fluid flows and wave phenomena. UK: Cambridge Univ. Press; 2013]. Although these compact schemes have been developed originally for direct simulation of incompressible flows, it is shown here that the same can be used for compressible flows, with shock-boundary layer interactions clearly captured for flow past NACA 0012 and NLF airfoils. Numerical higher order diffusion terms which are used for incompressible flows, have been replaced here by the pressure-based artificial diffusions proposed by Jameson et al. [Jameson A, Schmidt W, Turkel E. Numerical solution of the Euler equations by finite volume methods using Ruge-Kutta time stepping schemes. AIAA Paper 1981-1259. AIAA 14th fluid and plasma dynamics conference. Palo Alto, CA;. Such second and fourth order diffusion terms are used adaptively at selective points, located by the pressure switch. Developed computational methods used here are validated for cases with and without shocks, for which experimental results are available. Apart from surface pressure coefficient, contours of physical quantities are presented to explain the time-accurate results. Presented methods are robust and the results can be gainfully used to study shock formation, drag divergence and buffet onset of flow over airfoils.

29th International Symposium on Shock Waves 2, 2015

In this note, we relate the two well-known difficulties of Godunov schemes: the carbuncle phenomena in simulating high Mach number flow, and the inaccurate pressure profile in simulating low Mach number flow. We introduced two simple low-Mach-number modifications for the classical Roe flux to decrease the difference between the acoustic and advection contributions of the numerical dissipation. While the first modification increases the local numerical dissipation, the second decreases it. The numerical tests on the double-Mach reflection problem show that both modifications eliminate the kinked Mach stem suffered by the original flux. These results suggest that, other than insufficient numerical dissipation near the shock front, the carbuncle phenomena is strongly relevant to the non-comparable acoustic and advection contributions of the numerical dissipation produced by Godunov schemes due to the low Mach number effect.

Journal of Computational and Applied Mathematics, 2013

The proper scaling of the pressure-velocity coupling that arises from the Momentum Interpolation approach for unsteady calculation in low Mach number flow is first identified. Then, it is used to suggest a modification of the AUSM + -up scheme that allows acoustic simulations in low Mach number flow.

Computer Physics Communications, 2007

We study central-upwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707‐740]. Similarly to the staggered central schemes, these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution, simplicity, universality, and robustness. At the same time, the central-upwind framework

Inertia terms are inffoduced in a Godunoy-type scheme. The resulting scheme, called AUSM-IT, is designed as an extension of the AUSM+-up scheme, in order to allow full Mach number range calculations of unsteady flows. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the exact conservation of the disuete linear acoustic energy in the low Mach number limit. Its capability to propedy handle low Mach number unsteady flows, that may include acoustic wayes or discontinuities, is uumerically illustrated.

Computers & Fluids, 2006

The piece-wise parabolic method (PPM) is applied to simulations of forced isotropic turbulence with Mach numbers ∼ 0.1 . . . 1. The equation of state is dominated by the Fermi pressure of an electron-degenerate fluid. The dissipation in these simulations is of purely numerical origin. For the dimensionless mean rate of dissipation, we find values in agreement with known results from mostly incompressible turbulence simulations. The calculation of a Smagorinsky length corresponding to the rate of numerical dissipation supports the notion of the PPM supplying an implicit subgrid scale model. In the turbulence energy spectra of various flow realisations, we find the so-called bottleneck phenomenon, i. e., a flattening of the spectrum function near the wavenumber of maximal dissipation. The shape of the bottleneck peak in the compensated spectrum functions is comparable to what is found in turbulence simulations with hyperviscosity. Although the bottleneck effect reduces the range of nearly inertial length scales considerably, we are able to estimate the value of the Kolmogorov constant. For steady turbulence with a balance between energy injection and dissipation, it appears that C ≈ 1.7. However, a smaller value is found in the case of transonic turbulence with a large fraction of compressive components in the driving force. Moreover, we discuss length scales related to the dissipation, in particular, an effective numerical length scale ∆ eff , which can be regarded as the characteristic smoothing length of the implicit filter associated with the PPM.

2014

Inertia terms are inffoduced in a Godunoy-type scheme. The resulting scheme, called AUSM-IT, is designed as an extension of the AUSM+-up scheme, in order to allow full Mach number range calculations of unsteady flows. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the exact conservation of the disuete linear acoustic energy in the low Mach number limit. Its capability to propedy handle low Mach number unsteady flows, that may include acoustic wayes or discontinuities, is uumerically illustrated.

Final Report, 1992

NAqA- CR-191647 ONERA M6 wing, and (5) unsteadyflow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasisof the test caseswas validation of code,and assessment of performance, as well as demonstrationof flexibility.

The problem of accurately computing low Mach number flows, with the specific intent of studying the interaction of sound waves with incompressible flow structures, such as concentrations of vorticity is considered. This is a multiple time (and/or space) scales problem, leading to various difficulties in the design of numerical methods. Concentration is on one of these difficulties - the development of boundary conditions at artificial boundaries which allow sound waves and vortices to radiate to the far field. Nonlinear model equations are derived based on assumptions about the scaling of the variables. Then these are linearized about a uniform flow and exact boundary conditions are systematically derived using transform methods. Finally, useful approximations to the exact conditions which are valid for small Mach number and small viscosity are computed.