MATHEMATICAL AND NUMERICAL FORMULATION

A complete description of the hybrid theory behind the present hybrid approach is given in AIAA-99-0042 (Xu and Sankar, 1999). Sankar and his coworkers have also used the hybrid procedure in several fixed and rotary wing solvers.  For this reason, only the turbulence and transition model enhancements, and the extension of the flow solver to yaw conditions are presented here. 

Spalart-Allmaras Turbulence Model: 

The first enhancement to the hybrid analysis was the replacement of a simple algebraic eddy viscosity model with a phenomenological one-equation eddy viscosity model called the Spalart-Allmaras model.

In this model, the Reynolds stresses are given by 

  (1)

Where 

  (2)

 The eddy viscosity   is given by

  (3)

Where,

  And,  

Here   is the molecular viscosity. The quantity   is the working variable and obeys the transport equation.

  (4)

Here S is the magnitude of the vorticity, and

  (5)

Also, d is the distance to the closest wall, and,

  (6)

The function   is given by the following expression:

  (7)

Where

  (8)

And

  (9)

For large values of r, fw asymptotically reaches a constant value; therefore, large values of r can be truncated to 10 or so.

The Wall boundary condition is  = 0. In the freestream  = 0 is found to work best, provided numerical errors do not push   to negative values near the edge of the boundary layer.  Values below   are acceptable. 

The Spalart-Allmaras model has a built-in provision for driving the eddy viscosity to zero upstream of the transition point. This is done by the ft2 function, which goes to unity upstream of the transition point. 

  (10)

The trip function  is computed as follows. Let dt be the distance from the field point to the trip location, which is on a wall. Let the quantity   be the wall vorticity at the trip location, and   the difference between the velocity at the field point and that at the trip. Then one can compute an intermediate quantity  where  is the grid spacing along the wall at the trip location. Finally,

  (11)

The constants are:

cb1=0.1355,  =2/3, cb2= 0.622,  =0.41,

 , =0.3,  =2,  =7.1,  ,   , .

Further details on the Spalart-Allmaras model are given by Shur et al. (1998).

Eppler Transition Prediction Model 

The second enhancement to the hybrid method was the incorporation of transition line prediction models. Two models, one by Eppler, and the second by Michel have been investigated.

The Eppler’s transition model was implemented in the hybrid Navier-Stokes/Potential flow analysis, in the following manner. Every 10 time steps or so, the surface pressure distribution on the turbine blade is passed to an integral boundary layer analysis, one radial location at a time.  Inside the boundary layer analysis, the streamwise growth of laminar boundary layer quantities such as the momentum thickness , shape factor H, energy thickness 3, and the factor H32=3/ are computed using Thwaites' method (Thwaites, 1949). Transition is predicted to occur if the Reynolds number based on the momentum thickness becomes large so that: