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Computational solution of turbulent boundary layers
The simplest approach is based on the turbulent boundary-layer equations (7.108a, b) written in the form:
du dv д^с + ду=°
_дй _дй dp д / дй – р-л йд-х + ~Уд-у = ~Тх + д-у^д-у-^)
No attempt is made to write these equations in terms of non-dimensional variables as was done for Eqns (7.131) and (7.132). A similar procedure would be advantageous for computational solutions, but it is not necessary for the account given here.
The primary problem with solving Eqns (7.131) and (7.132) is not computational. Rather, it is that there are only two equations, but three dependent variables to determine by calculation, namely m, v and mV. The ‘solution’ described in Section 7.10 is to introduce an eddy-viscosity model (see Eqn 7.109) whereby
— du ~pu V = рєг — dy
In order to solve Eqns (7.144) and (7.145), єт must be expressed in terms of м (and, possibly, v). This can only be done semi-empirically. Below in Section 7.11.5 it will be explained how a suitable semi-empirical model for the eddy viscosity can be developed for computational calculation of turbulent boundary layers. First, a brief exposition of the wider aspects of this so-called turbulence modelling approach is given.
From the time in the 1950s when computers first began to be used by engineers, there has been a quest to develop increasingly more effective methods for computational calculation of turbulent flows. For the past two decades it has even been possible to carry out direct numerical simulations (DNS) of the full unsteady, threedimensional, Navier-Stokes equations for relatively simple turbulent flows at comparatively low Reynolds numbers.[51] Despite the enormous advances in computer power, however, it is unlikely that DNS will be feasible, or even possible, for most engineering applications within the foreseeable future.* All alternative computational methods rely heavily on semi-empirical approaches known collectively as turbulent modelling. The modern methods are based on deriving additional transport equations from the Navier-Stokes equations for quantities like the various components of the Reynolds stress tensor (see Eqn 7.107), the turbulence kinetic energy, and the viscous dissipation rate. In a sense, such approaches are based on an unattainable goal, because each new equation that is derived contains ever more unknown quantities, so that the number of dependent variables always grows faster than the number of equations. As a consequence, an increasing number of semi-empirical formulae is required. Nevertheless, despite their evident drawbacks, the computational methods based on turbulence modelling have become an indispensable tool in modern
engineering. A brief account of one of the most widely used of these methods will be given in Section 7.11.6.
Increasingly, an alternative approach to this type of turbulence modelling is becoming a viable computational tool for engineering applications. This is large – eddy simulation (LES) that was first developed by meteorologists. It still relies on semi-empirical turbulence modelling, however. A brief exposition will be given in Section 7.11.7.
