======Constant-Pressure Boundary Layer======
=====DNS by Spalart=====

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====Description====

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 10<sup>7</sup> 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.
        
<imgcaption cas33-geom| Flow geometry and configuration>{{ figs:case033:bl-geom.png | Flow configuration}}</imgcaption>

====Simulation Details====

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.

====Available Results====

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

[[case033-plots|Sample plots]] of selected quantities are available.

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

  * {{cdata:case033:cpbl-allfiles.zip|cpbl-allfiles.zip}}
  * {{cdata:case033:cpbl-allfiles.tar.gz|cpbl-allfiles.tar.gz}}

^  \(Re_{\theta}\)   ^  Profiles and stress budgets  ^
| \(Re_{\theta}=300\)   |  {{cdata:case033:simul1.dat|simul1.dat}}  |
| \(Re_{\theta}=670\)   |  {{cdata:case033:simul2.dat|simul2.dat}}  |
| \(Re_{\theta}=1410\)  |  {{cdata:case033:simul3.dat|simul3.dat}}  |


====References====

  - Spalart, P.R. (1986). Numerical simulation of boundary layers: Part 1. Weak formulation and numerical method. //NASA TM-88222//.
  - Spalart, P.R. (1988). [[https://doi.org/10.1017/S0022112088000345|Direct simulation of a turbulent boundary layer up to \(R_{\theta}=1410\)]]. //J. Fluid Mech.//, Vol. 187, pp. 61-98.

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Indexed data:
<data dbcase semi_confined_flow>
case       : 033
title      : Constant-Pressure Boundary Layer
author*    : Spalart
year       : 1988
type       : DNS
flow_tag*  : 2d, 2dbl
</data>