The simple equations given in Section 19.2 for boundary-layer thickness and skin friction coefficient for a turbulent flow over a flat plate are simplified results that are heavily empirically based. Modern calculations of turbulent flows over arbitrarily – shaped bodies involve the solution of the continuity, momentum, and energy equations along with some model of the turbulence. The calculations are carried out by means of computational fluid dynamic techniques. Here we will discuss only one model of tubulence, the Baldwin-Lomax turbulence model, which has become popular over the past two decades. We emphasize that the following discussion is intended only to give you the flavor of what is meant by a turbulence model.
19.3.1 The Baldwin-Lomax Model
In order to include the effects of turbulence in any analysis or computation, it is first necessary to have a model for the turbulence itself. Turbulence modeling is a state – of-the-art subject, and a recent survey of such modeling as applied to computations is given in Reference 85. Again, it is beyond the scope of the present book to give a detailed presentation of various turbulence models; the reader is referred to the literature for such matters. Instead, we choose to discuss only one such model here, because: (a) it is a typical example of an engineering-oriented turbulence model, (b) it is the model used in the majority of modern applications in turbulent, subsonic, supersonic, and hypersonic flows, and (c) we will discuss in the next chapter several applications which use this model. The model is called the Baldwin-Lomax turbulence model, first proposed in Reference 86. It is in the class of what is called an “eddy viscosity” model, where the effects of turbulence in the governing viscous
flow equations (such as the boundary-layer equations or the Navier-Stokes equations) are included simply by adding an additional term to the transport coefficients. For example, in all our previous viscous flow equations, /л is replaced by (ц. + t±r) and к by (к + kj) where jij and kT are the eddy viscosity and eddy thermal conductivity, respectively—both due to turbulence. In these expressions, ji and к are denoted as the “molecular” viscosity and thermal conductivity, respectively. For example, the x momentum boundary-layer equation for turbulent flow is written as
Moreover, the Baldwin-Lomax model is also in the class of “algebraic,” or “zero – equation,” models meaning that the formulation of the turbulence model utilizes just algebraic relations involving the flow properties. This is in contrast to one – and two-equation models which involve partial differential equations for the convection, creation, and dissipation of the turbulent kinetic energy and (frequently) the local vorticity. (See Reference 87 for a concise description of such one – and two-equation turbulence models.)
The Baldwin-Lomax turbulence model is described below. We give just a “cookbook” prescription for the model; the motivation and justification for the model are described at length in Reference 86. This, like all other turbulence models, is highly empirical. The final justification for its use is that it yields reasonable results across a wide range of Mach numbers, from subsonic to hypersonic. The model assumes that the turbulent-boundary layer is split into two layers, an inner and an outer layer, with different expressions for jiT in each layer:
(Mr) inner У — ycrossover r 1
, , . [19.6]
‘ M ‘/’ Jouter У _ ycrossover
where у is the local normal distance from the wall, and the crossover point from the inner to the outer layer is denoted by ycrossover- By definition, ycrossover is that point in the turbulent boundary where (мг)outer becomes less than (mt)inner – F°r the inner region:
and к and А+ are two dimensionless constants, specified later. In Equation (19.7), m is the local vorticity, defined for a two dimensional flow as
du dv З у dx
For the outer region:
where К and Ccp are two additional constants, and Fwake and /’кіеь are related to the function
Equation (19.12) will have a maximum value along a given normal distance y; this maximum value and the location where it occurs are denoted by Fmax and ymax, respectively. ІП Equation (19.11), Fwake is taken to be either Углах Fmax ОҐ (2wk Углах Fj. j –
whichever is smaller, where Cwk is constant, and
t/dif = у/ U2 + V2
Also, in Equation (19.11), FKieb is the Klebanoff intermittency factor, given by
The six dimensionless constants which appear in the above equations are: A+ = 26.0, Ccp = 1.6, Скіеь = 0-3, Cwk = 0.25, к = 0.4, and К = 0.0168. These constants are taken directly from Reference 86 with the understanding that, while they are not precisely the correct constants for most flows in general, they have been used successfully for a number of different applications. Note that, unlike many algebraic eddy viscosity models which are based on a characteristic length, the Baldwin-Lomax model is based on the local vorticity a>. This is a distinct advantage for the analysis of flows without an obvious mixing length, such as separated flows. Note that, like all eddy-viscosity turbulent models, the value of iij obtained above is dependent on the flowfield properties themselves (for example ш and p); this is in contrast to the molecular viscosity ц. which is solely a property of the gas itself.
The molecular values of viscosity coefficient and thermal conductivity are related through the Prandtl number
In lieu of developing a detailed turbulence model for the turbulent thermal conductivity kT, the usual procedure is to define a “turbulent” Prandtl number as Pr7 = цтср/кт. Thus, analogous to Equation (19.15), we have
where the usual assumption is that Pr7 = 1. Therefore, /і 7 is obtained from a given eddy-viscosity model (such as the Baldwin-Lomax model), and the corresponding kT is obtained from Equation (19.16).
Turbulence itself is a flowfield; it is not a simple property of the gas. This is why, as mentioned above, in an algebraic eddy viscosity model the values of /x7 and kT depend on the solution of the flow field—they are not pure properties of the gas as are /г and k. This is clearly seen in the Baldwin-Lomax model via Equation (19.7),
where fi t is a function of the local vorticity in the flow, oj—a flow field variable which comes out as part of the solution for the particular case at hand.