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# [SCG48-32] Solution Orbit for k and epsilon Model for Turbulent Flow in Porous Media

Keywords:porous media, k-epsilon model, solution orbit

Turbulent flow within porous media such as fault rocks can affect dynamic earthquake rupture processes and has attracted interests of many researchers. Among models treating the turbulent flow, the k-epsilon model has widely been employed, where k is the turbulent kinetic energy and epsilon is the dissipation rate of k. However, analytical treatment associated with the effect of the initial values of k and epsilon on the final state has not been performed.

We assume a one-dimensional homogeneous porous medium and isotropic turbulence. We also assume that the density of the fluid, rho, and the porosity, phi, are constant, and the ensemble averages of phi k and phi epsilon will be written as k and epsilon below, respectively. With these assumptions, it should be emphasized that exactly the same straight line given by the equation epsilon=c

Actually, the analytical form of solution orbits is given by epsilon=epsilon

Note that k and epsilon vanish with the limit t \to \infty for usual isotropic turbulent flow, even though the solution orbit for such flow is the same as one obtained here. Actually, the usual turbulent flow is described by the limit

We assume a one-dimensional homogeneous porous medium and isotropic turbulence. We also assume that the density of the fluid, rho, and the porosity, phi, are constant, and the ensemble averages of phi k and phi epsilon will be written as k and epsilon below, respectively. With these assumptions, it should be emphasized that exactly the same straight line given by the equation epsilon=c

_{k}u_{D}k /\sqrt{*K*} is a nullcline common to both variables on the k-epsilon phase space, where c_{k}is a constant number, u_{D}is averaged Darcy velocity, and*K*is permeability. We can regard this line as a line attractor or repeller as observed in other systems such as Suzuki (2017). Moreover, the straight line epsilon=0 (k axis) is also a nullcline for epsilon.Actually, the analytical form of solution orbits is given by epsilon=epsilon

_{0}(k/k_{0})^C_{2}, where C_{2}is a constant, 1.9. Using this, we show below that the common nullcline epsilon=c_{k}phi u_{D}k/\sqrt{*K*} is a line attractor for this solution orbit. First, we define Region I as the region 0< epsilon < c_{k}u_{D}k /\sqrt{*K*} on the phase space, and Region II as the region epsilon > c_{k}u_{D}k/\sqrt{K}>0. We also define the point (k_{f}, epsilon_{f}) as the point where the solution orbit passing the point (k_{0}, epsilon_{0}) crosses the nullcline. With these definitions, we have relations k_{0}f and epsilon_{0}f if (k_{0}, epsilon_{0}) is in the Region I. This occurs because epsilon is proportional to k on the nullcline, while epsilon is proportional to k^C_{2}on the solution orbit and C_{2}>1. We can also conclude k_{0}>k_{f}and epsilon_{0}>epsilon_{f}if (k_{0}, epsilon_{0}) is in the Region II. Second, we should emphasize that k and epsilon increase (decrease) with increasing time in the Region I (II), since \partial k/\partial t and \partial epsilon/\partial t are positive (negative). Therefore, if (k_{0}, epsilon_{0}) is in the Region I (II), the solution moves to the upper right (lower left), and is absorbed into the nullcline with the limit t \to \infty. We can conclude that the nullcline is a line attractor, not a repeller. The steady stable solution is given by (k, epsilon)=(k_{f}, epsilon_{f}).Note that k and epsilon vanish with the limit t \to \infty for usual isotropic turbulent flow, even though the solution orbit for such flow is the same as one obtained here. Actually, the usual turbulent flow is described by the limit

*K*\to \infty in the present model. The nullcline is k axis in such a case, and the Region I vanishes. Therefore, all the solutions are absorbed into the origin. The finite*K*enables the turbulence to be survive with t \to \infty for the homogeneous state.