Fig. 3 Comparison between prediction and data of Refs. 4 and 3 for eddy viscosity.where (7b) Thus L is proportional to n v , increasing as the backflow develops and decreasing toward reattachment, in agreement with observations a and b.An eddy viscosity formula is now obtained from the following argument. As pointed out in Ref. 5, the basic relation v t = C^k 2 /e must be modified within the viscous sublayer to preserve the correct behavior v t~n 3 at the wall, giving rise to the formula v t = C(n)k 2 /(en + ). In a similar manner, eddy viscosity within the backflow is assumed to be of the form v t =f(n)k 2 /e. Using Eqs. (1-5), and (7) yields (8) =-(C*/2) 9/5 , B= (q/2) 3 / 5 -A where f(n/n b ) = A (n/n b ) This form of /seems to best correlate with the data. 3 ' 4 The corresponding formula within the viscous sublayer is of the form (^/Osub = c ( n ) [k 2 /ve] sub , where k is given by Eq. (6) and e = kl /2 /n v . Here, C(n b ) =C; l/4 (A + £)/2V2, A and B being given previously.This concludes the formulation of the model, which ignores the viscous region adjacent to the wall within the separation bubble. This is justifiable in view of observation c.
Testing of ModelIn order to test the model, arbitrary streamwise locations were selected from Ref. 4, flow C, x = 92, and from Ref. 3, x=\44.9 in. The two sets of data pertain to different geometries, initial conditions, and means by which flow separation was imposed. A large separation bubble existed in the flow of Ref. 4, and the chosen location is approximately in the middle of it. Figure 2 compares k from Eqs. ( 1) and (6), with the data. The agreement supports the Gaussian concept. Furthermore, calculated values of k b at this and other locations were found to be within only 2% of those obtained from the measurements. Based on the measured velocity profile and shear stress distribution at the selected station, eddy-viscosity data were compared with Eq.(8), as shown in Fig. 3. Since the velocity profile had to be numerically treated to derive du/dy, some error was introduced into the data points; nevertheless, the agreement is reasonable. In the case of Ref. 3, eddy viscosity was already given in data form. Figure 3 also shows comparison of v t prediction with these data at the selected location. Agreement is very good, even well beyond the backflow. In addition, predicted v t levels dropped about 45% when going from well upstream of separation into the bubble along a constant /?, as experimentally observed. 3 This leads credibility to the length scale L. These tests serve as a preliminary validation of the model.A parametric study showed that c/> = 0.5 is the best choice; this value is adopted as a constant of the model.
ConclusionsA k-e formulation has been developed for turbulence modeling within wall-bounded two-dimensional separation bubbles. The model is based on experimental obsevations. Two basic features of the model are: 1) turbulence kinetic energy within the backflow region is a Gaussian function of the distance from the wall; 2) the length scal...
