Constant-Pressure Boundary Layer

The turbulent boundary layer on a flat plate, with zero pressure gradient, was simulated numerically at four stations between \(Re_{\theta} = 225\) to \(Re_{\theta} = 1410\). 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.

 Flow configuration Fig. 1: Flow geometry and configuration

The numerical method is described in detail by Spalart (1986). 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 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, \(\delta^*\), 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, \(\delta\), or the momentum thickness, \(\theta\). The lengthscale \(y_o\) of the exponential mapping (Spalart 1986) is also kept at a constant multiple of \(\delta^*\). 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.

Governing Equations

The solution was a set of equations which, when solved with periodic conditions 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.

Data is available at three momentum thicknesses, \(Re_{\theta}=300\), \(670\) and \(1410\), consisting of:

  • Profiles across the boundary layer of mean velocity and Reynolds stresses
  • Budgets of Reynolds stresses and turbulent kinetic energy

Sample plots of selected quantities are available.

The data can be downloaded as compressed archives from the links below, or as individual files.

\(Re_{\theta}\) Profiles and stress budgets
\(Re_{\theta}=300\) simul1.dat
\(Re_{\theta}=670\) simul2.dat
\(Re_{\theta}=1410\) simul3.dat
  1. Spalart, P.R. (1986). Numerical simulation of boundary layers: Part 1. Weak formulation and numerical method. NASA TM-88222.
  2. Spalart, P.R. (1988). Direct simulation of a turbulent boundary layer up to \(R_{\theta}=1410\). J. Fluid Mech., Vol. 187, pp. 61-98.

Indexed data:

case033 (dbcase, semi_confined_flow)
titleConstant-Pressure Boundary Layer
flow_tag2d, 2dbl