Primitive variable, strongly implicit calculation procedure for viscous flows at all speeds

Abstract

A coupled solution procedure is described for solving the compressible form of the time-dependent, twodimensional Navier-Stokes equations in body-fitted curvilinear coordinates. This approach employs the strong conservation form of the governing equations but uses primitive variables (M, v, p, T) rather than the more traditional conservative variables (p, pw, pv, e t ) as unknowns. A coupled modified strongly implicit procedure (CMSIP) is used to efficiently solve the Newton-linearized algebraic equations. It appears that this procedure is effective for Mach numbers ranging from the incompressible limit (M x ~ 0.01) to supersonic. Generally, smoothing was not needed to control spatial oscillations in pressure for subsonic flows despite the use of central differences. Dual-time stepping was found to further accelerate convergence for steady flows. Sample calculations, including steady and unsteady low-Mach-number internal and external flows and a steady shock-boundarylayer interaction flow, illustrate the capability of the present solution algorithm.

Industrial airflows numerical simulation in ducts and devices using all-speed algorithm in structured meshes

“…The density variation effect is indistinguishable from errors arising from numerical truncation due to the algorithm spatial precision order, thus generating physically dubious results, as also discussed in [13]. It is noteworthy that the method presented herein is always a second order in space, and for Mach numbers around 0.1 the flow is typically incompressible.…”

“…γ ar = 1.4 (11) R = 287 J/kgK (12) C v = 718 J/kgK (13) T 0 = 302.53 K (14) T amb = 300.00 K (15) u jet = 43.80 m/s (16) T static / T 0 = P static / P 0 ( ) γ γ −1 (17) …”

Résolution numérique d'écoulements 3 dimensions avec une nouvelle méthode de volumes finis pour maillages non structurés /

“…Pour faciliter la résolution, une dérivée temporelle de la pression peut être ajoutée à l'équation de continuité (méthode de compressibilite artificielle). Chorin [24], en 1967, a été le précurseur de cette approche qui est mal-adaptée aux problèmes transitoires : une boucle devant être effectuée à chacun des pas de temps pour que la solution soit satisfaisante [23]. Les méthodes les plus populaires sont basées sur un découplage des équations et sont souvent appelées méthodes de correction de pression.…”

The Geometric Conservation Based Algorithms For Multi-Fluid Flow At All Speeds

“…When dealing with all-speed flows, pressurevelocity-density coupling has to be accounted for. Pressure-based algorithms have been extended successfully [19][20][21][22][23][24][25][26][27] to account for this additional coupling.…”

“…Some of these factors are directly related to the maturity of several numerical aspects at the core of CFD. These include: multi-grid acceleration techniques [1][2][3][4] with enhanced equation solvers [5,6] that have decreased the computational cost of tackling large problems, better discretization techniques, unstructured grids [7][8][9][10][11][12], bounded high resolution schemes [13][14][15][16][17][18], as well as improved pressure-velocity (and density) coupling algorithms for fluid flow at all speeds [19][20][21][22][23][24][25][26][27]. Other factors, independent of the CFD industry, have to do with the exponential increase in processor power and decrease in microprocessor cost, whereby multiprocessors systems with large memory can now be set up at a fraction the cost of the super-computers of a decade ago.…”