The turbulent boundary layer on a flat plate, with zero pressure gradient, was simulated numerically at four stations between = 225 to = 1410. The three-dimensional time-dependent Navier-Stokes equations are solved using a spectral method with up to about 107 grid points. Periodic spanwise and streamwise~ conditions are applied, and a multiple-scale procedure is applied to approximate the slow streamwise growth of the boundary layer.
The numerical method was described in detail by Spalart (1986a). It is fully spectral in space, based on Fourier series in the directions parallel to the plate and an exponential mapping with Jacobi polynomials in the normal, semi-infinite direction. The time integration is second-order accurate and hybrid; it uses a low-storage Runge-Kutta scheme (Wray 1987) for the transport term and the Crank-Nicolson scheme for the Stokes terms. If Reynolds-number effects are to be studied by numerical simulation, it is essential to ensure that the different cases are not run with (effectively) different resolution, which could induce spurious variations. A similar problem can occur in experiments, for instance if a probe of fixed size is used while an increase in the Reynolds number decreases the scales of the turbulence.
There are two aspects to the question of resolution. One is the size of the domain in the directions parallel to the wall (or equivalently the smallest wavenumber). It was decided to keep the ratio of these dimensions to the displacement thickness * the same in all the simulations. Thus if there is an effect of the confinement of the flow inside a finite domain, the effect will be as independent of Reynolds number as possible. The displacement thickness is an appropriate macroscale of the flow and, with the present method, happens to be easier to control than the boundary-layer thickness or the momentum thickness . The lengthscale yo of the exponential mapping (Spalart 1986 a) is also kept at a constant multiple of * The other aspect is of course the grid spacing (or equivalently the largest wavenumber). In this case the wall region is the most sensitive and it was decided that the grid spacing should be fixed, in wall units. Thus the effects of numerical truncation will be as independent of Reynolds number as possible.
The solution was a set of equations which, when solved with periodic in the streamwise (x) direction, can provide a good approximation to the local state of a boundary layer that has a slow spatial development. The incentives to use periodic conditions are both numerical (the high accuracy of Fourier series) and physical (no need to provide turbulent inflow conditions, improved statistical sample). The idea is to use the fact that both the thickness of the boundary layer and the energy level of the turbulence vary slowly as functions of x. The final product is a set of small 'growth terms' that are added to the usual Navier-Stokes equations.
Previous and Reference Numerical Solutions
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