Abstract
The artificial compressibility method is quite different from the pressure projection approach in both the nature of the formulation and the subsequent numerical algorithm. In an artificial compressibility method, a fictitious time derivative of pressure is added to the continuity equation so that the set of equations modified from the incompressible Navier-Stokes equations can be solved implicitly by marching in pseudo time. When a steady-state solution is reached, the original equations are recovered. To obtain time accuracy, an iterative technique can be employed at each time level, which is equivalent to solving the governing equations for steady state at each time level. Using a large, artificial compressibility parameter to spread artificial waves quickly throughout the computational domain, and allowing some residual level of the mass conservation equation, the computing time requirement for time accurate solutions may be controlled within approximately one order-of-magnitude higher than the steady-state computations. In the artificial compressibility approach, the mass conservation does not have to be strictly enforced at each time step, and this gives robustness during iteration.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Armaly, B. F., Durst, F., Pereira, J. C. F., Schonung, B.: Experimental and theoretical investigation of backward facing step flow. J. Fluid Mech., 127, 473–496 (1983)
Beam, R. M., Warming, R. F.: An implicit factored scheme for the compressible Navier-Stokes equations. AIAA J., 16, 393–402 (1978)
Briley, W. R., McDonald, H.: Solution of the multidimensional compressible Navier-Stokes equations by a generalized implicit method. J. Comp. Phys., 24, No. 4, 372–397 (1977)
Harten, A., Lax, P. D., Van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25, No. 1, 35 (1983)
Housman, J., Kiris, C., Hafez, M. Preconditioned methods for simulations of low speed compressible flows. Comp. Fluids, 38, 7, 1411–1423 (2009)
Jameson, A., Yoon, S.: Multigrid solution of the Euler equations using implicit schemes. AIAA J., 24, 1737–1743 (1986)
Kreiss, H. O.: On difference approximations of the dissipative type for hyperbolic differential equation. Comm. Pure Appl. Math., 17, 335–353 (1964)
Merkle, C. L.: Preconditioning methods for viscous flow calculations. In Computational Fluid Dynamics Review 1995, ed. by Hafez, M. and Oshima, K., Wiley, New York (1995)
Pulliam, T. H.: Artificial dissipation models for the Euler equations. AIAA J., 24, 1931–1940 (1986)
Pulliam, T. H., Chaussee, D. S.: A diagonal form of an implicit approximate-factorization algorithm. J. Comput. Phys., 39, 347–363 (1981)
Roe, P. L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43, 357 (1981)
Rogers, S. E., Chang, J. L. C., Kwak, D.: A diagonal algorithm for the method of pseudocompressibility. J. Comput. Phys., 73, No. 2, 364–379 (1987)
Rogers, S. E., Kwak, D.: An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J., 28, No. 2, 253–262 (1990) (Also, AIAA Paper 88-2583, 1988)
Steger, J. L., Kutler, P.: Implicit finite-difference procedures for the computation of vortex wakes. AIAA J., 15, No. 4, 581–590 (1977)
Turkel, E.: Symmetrization of the fluid dynamic matrices with applications. Math. Comput., 27, 729–736 (1973)
Venkataswaran, S., Merkle, C. L.: Evolution of artificial compressibility methods in CFD. In Numerical Simulations of Incompressible Flows, ed. by Hafez, M., World Scientific, Singapore (2002)
Warming, R. F., Beam, R. M., Hyett, B. J.: Diagonalization and simultaneous symmetrization of the gas-dynamic matrices. Math. Comput., 29, 1037–1045 (1975)
Yee, H. C.: Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations. Comp. Math. Appl., 12A, Nos. 4/5, 413–432 (1986)
Rudy, D. H., Strikwerda, J. C.: A nonreflecting outflow boundary condition for subsonic Navier-Stokes calculations. J. Comp. Phys, 36, 55–70 (1980)
Steger, J. L., Warming, R. F.: Flux vector splitting of the inviscid gasdynamic equations with application to finite difference methods. J. Comp. Phys., 40, No. 2, 263–293 (1981)
Barth, T. J.: Analysis of implicit local linearization techniques for upwind and TVD algorithms. AIAA Paper 87-0595 (1987)
Chakravarthy, S. R., Anderson, D. A., Salas, M. D.: The split-coefficient matrix method for hyperbolic systems of gas dynamics. AIAA Paper 80-0268 (1980)
Chakravarthy, S. R., Osher, S.: A new class of high accuracy TVD schemes for hyperbolic conservation laws. AIAA Paper 85-0363 (1985)
Chang, J. L. C., Kwak, D.: On the method of pseudo compressibility for numerically solving incompressible flows. AIAA Paper 84-0252 (1984)
Flores, J.: Convergence acceleration for a three-dimensional Euler/Navier-Stokes zonal approach. AIAA Paper 85-1495 (1985)
Hafez, M.: On the incompressible limit of compressible fluid flow. In Computational Fluid Dynamics for the 21st Century, ed. by Hafez, M. M., Morinishi, K., Periaux, J. and Satofuka, N., Springer (2001)
Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the Euler equations by finite volume methods using runge-kutta stepping scheme. AIAA Paper 81-1259 (1981)
MacCormack, R. W.: Current status of numerical solutions of the Navier-Stokes equations. AIAA Paper 85-0032 (1985)
Merkle, C. L., Tsai, P. Y. L.: Application of Runge-Kutta schemes to incompressible flows. AIAA Paper 86-0553 (1986)
Rai, M. M.: Navier-Stokes simulations of blade-vortex interaction using high-order accurate upwind schemes. AIAA Paper 87-0543 (1987)
Rapposelli, E., Cervone, A., Bramanti, C., d’Agostino, L.: Thermal cavitation experiments on a NACA 0015 hydrofoil. Proceeding of the FEDSM’03 4th ASME/JSME Joint Fluids Engineering Conference, Honolulu, Hawaii, July (2003)
Rogers, S. E., Kwak, D.: Numerical solution of the incompressible Navier-Stokes equations for steady and time-dependent problems. AIAA Paper 89-0463 (1989)
Salvetti, M.-V., Beux, F.: Liquid flow around non-cavitating and cavitating NACA 0015 hydrofoil. Mathematical and Numerical Aspects of Low Mach Number Flows, Porquerolles, France, Workshop Problem (2004)
Yoon, S., Kwak, D.: Artificial dissipation models for hypersonic external flow. AIAA Paper 88-3708 (1988)
Yoon, S., Kwak, D.: LU-SGS implicit algorithm for three-dimensional incompressible Navier-Stokes equations with source term. AIAA Paper 89-1964 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Kwak, D., Kiris, C.C. (2011). Artificial Compressibility Method. In: Computation of Viscous Incompressible Flows. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0193-9_4
Download citation
DOI: https://doi.org/10.1007/978-94-007-0193-9_4
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0192-2
Online ISBN: 978-94-007-0193-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)