Objectives_template

Module 8:

Lecture 27: Mathematical Modeling of Turbulent Flows (Continued)

One-Equation Model

This is a very popular model used widely for external aerodynamic flows. Spalart and Allmaras (1992) have developed one equation model, which is designed specifically for aerospace applications involving wall-bounded flows. The Reynolds stresses are given by

(27.1)

The eddy viscosity v_t is given by

(27.2)

is the molecular viscosity, obeys the transport equation


	(27.3)

Here

(27.4)

where S is the magnitude of the vorticity, and d is the distance to the closest wall. The function f_w is

(27.5)

For large r, f_w reaches a constant, so large values of r can be truncated to 10 or so. The wall boundary condition is . In the freestream 0 is best, provided numerical errors do not push to negative values near the edge of the boundary layer (the exact solution cannot go negative). Values below /10 will be acceptable; the same applies to the initial condition.

In some codes a portion of the solid surface, typically the fuselage, is treated with a free-slip condition while another portion, typically the wing, is treated with a no-slip condition. For , the appropriate condition on the free-slip surface is a Neumann condition (zero normal derivative). In addition, the free-slip wall points are not included in the search when d is computed for the field points.

The function f_t2 is

(27.6)

The trip function f_t1 is as follows. The quantity d_t is the distance from the field point to the trip, which is on a wall, w_t is the wall vorticity at the trip, and ΔU is the difference between the velocity at the field point and that at the trip. Then g_t = min (0.1, ΔU / w_tδ x_t where δx_t is the grid spacing along the wall at trip, and

(27.7)

The constants are c_b1 = 0.1355, σ = 2/3, c_b2 = 0.622, = 0.41, c_w₁ = c_b1/k² + (1 + c_b2 )/σ , c_w₂ = 0.3, c_w₃ = 2, c_v₁ = 7.1, c_t₁ = 1, c_t2 = 2, c_t3 = 1.2 and c_t4 = 0.5.

The model has demonstrated good results for boundary layer flows subject to adverse pressure gradients. It performs well in the near wake and appears to be a good candidate for more complex flows.