Direct Simulations of Periodic Burgers’ Flow

Abstract

It is a common practice in turbulence research to make use of available experimental information to test analytical theories or closure models. An attractive option in this regard consists of computing numerical solutions of the Navier-Stokes equations, with grid sizes and time steps so small that all the scales present in the fluid motion are well resolved. This is what is now known as “direct simulations of turbulence.” One of the advantages of this procedure is that more precise control can be exerted on “numerical,” as opposed to laboratory, experiments. On the other hand, as was mentioned earlier, the main drawback of direct simulations is that the number of nodes required for the resolution of the smallest scales of the motion is too large for the range of Reyolds numbers of interest in practice (see expression (2.1)). Moreover, even for low Reynolds numbers, for which such simulations are feasible, required supercomputer (e.g., CRAY-1, CRAY-XMP) CPU-time is of the order of several tenths of hours. For these reasons, especially in the initial testing stages of filtering techniques, a one-dimensional analogue of the (three-dimensional) Navier-Stokes equations, namely, Burgers’ equation, has been frequently used. Due to the one-dimensional character of this equation, the resolution requirement for “turbulence” in this context is much less restrictive than in the Navier-Stokes case. Accordingly, direct simulations with Burgers’ equation can be performed at much smaller computer cost. In the next section we define the model problem that we used to test the filtering theory that was previously presented.