SPACE-TIME STRUCTURE AROUND THE TRANSITION POINT IN CHANNEL FLOW REVEALED BY THE STOCHASTIC DETERMINISM

The transition to turbulence (Reynolds, 1883) has attracted people. Large eddy simulation (LES) and direct numerical simulation (DNS) of the transition to turbulence in straight channels employed the spatial cyclic boundary conditions between the inlet and outlet of the channel. (Moin and Kim, 1982; Kawamura and Kuwahara, 1985) Thus, these previous researches capture only the transition in time, although the spatial transition point where the laminar flow changes to turbulence could not be computed. Recently, some approaches tried to compute the transition point in straight channel for the flows having large disturbances at the inlet. (Oida and Kuwahara, 2003) However, computations of the transition points in the flows with various inlet-disturbances and also for various Reynolds numbers are still in an infant stage, although it is necessary to predict the transition points for airfoil optimizations and micro-fluids such as blood and fuel cell. Thus, we proposed the method called “stochastic determinism”, based on the deterministic Navier-Stokes equation and stochastic artificial disturbances. (Naitoh et al., 2008) Here, we show the method in detail and also clarify the space-time structure after an impulsive start for a wide range of Reynolds numbers. STOCHASTIC DETERMINISM Convergence criterion for the matrix calculation in computational fluid dynamics has been based on mathematics until now. Roache, 1972. However, the numerical errors should be used adequately on the basis of physical evaluation, because we cannot simulate the unstable asymmetric flows such as Karman vortex streets without numerical disturbances. Asymmetric distribution of numerical errors in space brings the up-and-down asymmetric flows, although people have not mentioned explicitly. We should find the appropriate values of the numerical errors in the analytical domain, which correspond to the physical fluctuations. We propose the physical criterion of convergence, appearing after scales falling from our eyes, in which the level of numerical truncation errors is related to actual velocity fluctuation at the inlet. Combination of this physical criterion and the finite difference method extended based on the multi-level formulation (Naitoh and Kuwahara, 1992), which can calculate spatial derivatives of physical quantities and integrated quantities accurately, leads us to the new stage of computational fluid dynamics. Starting point of the methodology is related to the equation of the divergence of velocity in the multi-level formulation, which is transformed from the compressible Navier-Stokes equation while maintaining the Gibbs formula. (Naitoh and Kuwahara, 1992.) The formulation shows that the second derivative of velocity, the divergence of velocity, controls the physical quantities such as pressure, velocity, and temperature. Thus, in the following sections, we will focus on the control of the numerical error of the divergence of velocity. NUMERICAL METHOD The three-dimensional incompressible Navier-Stokes equation in non-conservative form is employed as the governing equation (Eq. 1).