Navier-Stokes Equation and the Transition Model

The constant property Navier-Stokes equations adequately model the fluid physics needed to perform practical laminar – and turbulent-flow computations in the Reynolds number range typically used by the low Reynolds number flyers:

^ = 0,

д X:

д U: d 1 d p d2

—- + —— (U:U j ) =——– + V (U: ),

dt d Xj 1 p d X : дX2-

where u: are the mean flow velocities and v is the kinematic viscosity. For turbulent flows, turbulent closures are needed if one is solving the ensemble-averaged form of the Navier-Stokes equations. Numerous closure models have been proposed in the literature [102]. Here we present the two-equation к — ю turbulence model [102] as an example. For clarity, the turbulence model is written in Cartesian coordinates as follows:

For the preceding equations, k is the turbulent kinetic energy, m is the dissipation rate, vT is the turbulent kinematic eddy viscosity, ReT is the turbulent Reynolds number, and а0, в, R^, Rk, and Rm are model constants. To solve for the transition from laminar to turbulent flow, the incompressible Navier-Stokes equations are coupled with a transition model.

The onset of laminar-turbulent transition is sensitive to a wide variety of disturbances, as produced by the pressure gradient, wall roughness, free-stream turbulence, acoustic noise, and thermal environment. A comprehensive transition model considering all these factors currently does not exist. Even if we limit our focus to free-stream turbulence, it is still a challenge to provide an accurate mathe­matical description. Overall, approaches to transition prediction can be categorized as (i) empirical methods and those based on linear stability analysis, such as the eN method [96]; (ii) linear or non-linear parabolized stability equations [103]; and (iii) large-eddy simulation (LES) [104] or direct numerical simulation (DNS) methods [105].

Empirical methods have also been proposed to predict transition in a sepa­ration bubble. For example, Roberts [93] and Volino and Bohl [106] developed models based on local turbulence levels; Mayle [107], Praisner and Clark [108], and Roberts and Yaras [109] tested concepts by using the local Reynolds num­ber based on the momentum thickness. These models use only one or two local parameters to predict the transition points and hence often oversimplify the down­stream factors such as the pressure gradient, surface geometry, and surface rough­ness. For attached flow Wazzan et al. [110] proposed a model based on the shape factor H. Their model gives a unified correlation between the transition point and Reynolds number for a variety of problems. For separated flow, however, no similar models exist, in part because of the difficulty in estimating the shape factor.

Among the approaches employing linear stability analysis, the eN method has been widely adopted [111] [112]. It solves the Orr-Sommerfeld equation to evaluate the local growth rate of unstable waves based on velocity and temperature profiles over a solid surface. Its successful application is exemplified in the popularity of airfoil analysis software such as XFOIL [96]. XFOIL uses the steady Euler equa­tions to represent the inviscid flows, a two-equation integral formulation based on dissipation closure to represent boundary layers and wakes, and the eN method to
tackle transition. Coupling the Reynolds-averaged Navier-Stokes (RANS) solver with the eN method to predict transition has been practiced by Radespiel et al. [113], Stock and Haase [114], and He et al. [115]. An application of this approach for low Reynolds number applications can be found in the work of Yuan et al. [116] and Lian and Shyy [117].

The eN method is based on the following assumptions: (i) the velocity and temperature profiles are essentially 2D and steady, (ii) the initial disturbance is infinitesimal, and (iii) the boundary layer is thin. Even though in practice the eN method has been extended to study 3D flow, strictly speaking, such flows do not meet the preceding conditions. Furthermore, even in 2D flow, not all these assumptions can be satisfied [118]. Nevertheless, the eN method remains a practical and useful approach for engineering applications.

Advancements in turbulence modeling have made possible alternative approaches for transition prediction. For example, Wilcox devised a low Reynolds number к – ю turbulence model to predict transition [119]. One of his objectives is to match the minimum critical Reynolds number beyond which the TS wave begins forming in the Blasius boundary-layer context. However, this model fails if the separation-induced transition occurs before the minimum Reynolds number is reached, as frequently occurs in the separation-induced transition. Holloway et al. [120] used unsteady RANS equations to study the flow separation over a blunt body for the Reynolds number range of 104 to 107. It has been observed that the predicted transition point can be too early even for a flat-plate flow case, as illustrated by Dick and Steelant [121]. In addition, Dick and Steelant [122] and Suzen and Huang [123] incorporated the concept of an intermittency factor to model transitional flows. One can model this either by using conditional-averaged Navier-Stokes equations or by multiplying the eddy viscosity by the intermittency factor. In either approach, the intermittency factor is solved based on a transport equation, aided by empirical cor­relations. Mary and Sagaut [124] studied the near-stall phenomena around an airfoil using LES, and Yuan et al. [116] studied transition over a low Reynolds number airfoil using LES.