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Two-equation methods
Thwaites’s method and White’s equilibrium method are examples of one-equation integral methods, meaning that they integrate one differential equation to obtain the solution. One of their main drawbacks is that they cannot correctly represent the behavior of a separated boundary layer. For Thwaites’s method this can be seen by examining the H and A column values in Table 4.1. For adverse pressure gradients (A < 0) this
true H(A) function is actually two-valued, with one H < 4 value which is the attached solution, and another H > 4 value which is the separated solution. Even if the curve-fit H(A) expression (4.85) were somehow modified to have this two-value form, it would be impossible to use in the H evaluation step (4.90), since there’s no way to know whether to choose the attached or the separated H value for any given negative A value.
This problem is eliminated by the so-called two-equation integral methods, such as those of LeBalleur [23], Whitfield et al [24], and Drela et al [6]. These methods integrate both the von Karman equation (4.28) for 9(s), and also the kinetic energy equation (4.35) for в*, or equivalently for H* = 0*/0. The latter is actually obtained more conveniently from the combination [equation (4.35) ]/в* — [ equation (4.28) ]/в which produces the kinetic energy shape parameter equation.
1 dH* = 2cv cf ґ 2H** в due
H* ds H* 2 H* ) ue ds
Two equation methods assume that H and H* are uniquely related via a H*(H) correlation function, so that equation (4.98) above is in effect an ODE for H(s). For laminar flow, the H*(H) function is implied by Table 4.1. For turbulent flow, a H*(H, Bee) function is obtained from the self-similar turbulent profiles shown in Figure 4.14, but actually differs very little from the laminar version. Since H is calculated directly in the two-equation methods, there is no ambiguity as to whether the flow is attached or separated at any given location. In these methods A or Л is not needed and is not used explicitly.
Another type of two-equation method is developed by Head [25] and Green et al [26], and is based on the entrainment equation, which is an integral form of the mass equation. The behavior of entrainment-based methods is similar to those of the kinetic energy-based methods, and the details are not important here.
Besides enabling the representation of a separated boundary layer, two-equation methods are considerably more accurate than the one-equation methods, especially for turbulent flow. Since their derivation makes the same basic correlation assumptions as the one-equation methods, i. e. the Falkner-Skan solutions for laminar flow and the equilibrium profiles and G-beta locus for turbulent flow, presenting them in detail here would add little besides complexity. The reader is referred to references for the derivation details.
