Numerical simulation of a developed shear turbulence

Abstract

Results of two-dimensional numerical studies of turbulence that arises at the interface of twoProblem Statement. When two incompressible semi-infinite streams of a fluid (or a gas) move in one direction with different velocities U1 and U2, turbulence develops at the interface between them. This motion is self-similar; therefore, the thickness of the turbulent mixing zone can be represented aswhere c~, is a constant determined experimentally~ t is the time of interaction of the two flows, and f(Pl/P2) is a dimensionless function that depends oil the density difference and normalized so that f(1) = 1. The difference IU1 -U21 is an invariant of the Galilean transform, and the flow pattern remains unchanged on interchanging the two flow velocities: U1 ~-~ [72. This interchanging is equivalent to the case where the densities are interchanged: Pl ~-~ P2. Hence, the function f(x) possesses the following property:Since any experimental study in such a time statement of the problem of interest seems to be difficult to perform, the shear instability is usually studied within the following spatial formulation. Two incompressible fluids (or two gases) with densities Pl and P2 are considered. These fluids move along the semiplane y = 0, x < 0 that separates them, with velocities U1 and U2, respectively (UI > U2) (Fig.l). We define the Atwood number as A = (pl -P2)/(Pl + P2). In the case under consideration, this number can be either positive or negative. At the point x = 0, the fluids are in contact, while at x > 0, turbulence develops at the interface (region I in Fig. 1).The spatial problem can be substituted with a time one using the relation x = Uot. The physical meaning of the velocity Uo becomes clear from the following consideration. If, at t = 0, the observer is located at the point x ---0, y = 0, and, afterwards, he moves along the x-axis with velocity U0, then, for him, the development of the mixing zone is described by relation (1). Since~ at a time t > 0, the observer is situated at the point x = Uot, the width of the mixing zone can be represented as L = 0.5auf(pl/P2)lU1 -U21(x/Uo).