THE NUMERICAL SIMULATION OF 



VISCOUS INCOMPRESSIBLE 



FLUID FLOWS 



C. W. Hirt 



Los Alamos Scientific Laboratory 



University of California 



Los Alamos, New Mexico 



ABSTRACT 



This paper describes anEulerian finite -difference technique for solving 

 the complete nonlinear Navier-Stokes equations. The technique is 

 applicable to transient flows of viscous, incompressible fluids with free 

 surfaces. The basic features of the finite -difference method are first 

 explained in terms of a simple linear convection equation. The dis- 

 cussion covers questions of accuracy and computational stability. These 

 results are then applied to the solution of the complete time -dependent 

 Navier-Stokes equations. Techniques are also described for incorpo- 

 rating free -surface stress conditions, as well as other boundary condi- 

 tions. The complete numerical scheme that is developed is referred 

 to as the Marker -and-Cell (MAC) method (1). Two applications of 

 the MAC method are discussed in detail. The first application is to 

 the flow of water under a sluice gate. This example illustrates how 

 the MAC method is kept computationally stable. The second application 

 deals with the formation of a hydratilic jump. This example reveals 

 several important aspects of the numerical treatnnent of boundary con- 

 ditions. The hydraulic jump example also illustrates an attempt of the 

 numerical method to simulate fluid turbulence. This leads to a discus- 

 sion of a new method for obtaining the numerical solution of time- 

 dependent fully turbulent flows. 



INTRODUCTION 



The Marker-and-Cell (MAC) technique is an Eulerian finite-difference 

 method for solving the complete nonlinear Navier-Stokes equations (1). The 

 MAC method is applicable to transient flows of viscous, incompressible fluids 

 with free surfaces. Examples are the surge of water under a sluice gate, the 

 splashing of liquid drops, and the formation of hydraulic jumps. 



In this paper the method is presented for arbitrary two-dimensional flows. 

 Three-dimensional calculations are not treated here since they are impractical 

 with the speed and size of today's computers. Special considerations are given 

 to the conditions for computational stability and accuracy, and to the derivation 

 of boundary conditions, especially at free surfaces. Finally, a brief discussion 

 is presented of a new method for studying fully turbulent flows. 



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