**3. Two- and three-dimensional test cases for low to high speeds**

Some outstanding test cases that make a good platform for measuring novel transition model performances are simulated by the foregoing transition models. These cases cover a wide range of flows from low speed two-dimensional flat plate and airfoil test cases to three-dimensional wind turbine blade and aircraft wing test cases from low to high speeds.

#### **3.1 Low speed flat plate test cases**

Well-known benchmark experiments such as the Schubauer and Klebanoff natural transition flat plate experiment [29] and the ERCOFTAC T3 series flat plate experiments by Savill [33] are used. The T3 series flat plate experiments consist of three zero pressure flat plate cases (T3A, T3B, and T3A-) and five variable pressure flat plate cases (T3C1, T3C2, T3C3, T3C4, and T3C5), in which the pressure gradients are generated using an adjustable upper tunnel wall. In all ERCOFTAC T3 test cases, the free stream turbulence intensities vary between 0.1 and 6%. **Table 1** summarizes the upstream conditions of the Schubauer and Klebanoff and the ERCOFTAC T3 flat plate experiments.

**Figure 2** shows the numerical and experimental skin friction coefficients of the zero pressure gradient test cases of S&K, T3A, T3B and T3A-, respectively. The figures include numerical predictions of several researchers, including for instance Suzen and Huang [9], Langtry and Menter [11], Walters and Cokljat [14], Menter et al. [22], Nagapetyan and Agarwal [24], and Medida [31]. In the S&K calibration case, the B-C model displays a good agreement with the experiment for the transition onset point similar to other methods. For the T3A and T3B cases, the B-C model shows rather late transition onset, whereas the other models predict some early or late onset points. Specifically, Nagapetyan and Agarwal [24] show a very good agreement with the experiment as to the transition onset and rapid skin-friction

#### *Boundary Layer Flows - Theory, Applications and Numerical Methods*


#### **Table 1.**

*Inlet conditions for the flat plate test cases.*

rise characteristic. Finally, for the T3A- case, the B-C [20], Menter et al. [22], Walters and Cokljat [14], and Nagapetyan and Agarwal [24] display early transition onset points with rather rapid rise in skin-friction, whereas two-equation Langtry and Menter [11] and Medida [31] models show quite good onset point and a gradual rise in the skin friction.

**Figure 3** depicts numerical and experimental skin friction coefficients for the T3C series variable pressure flat plate test cases. The T3C series flat plate test cases represent actual turbine characteristics by changing the pressure gradient by changing the upper wall profile of the wind tunnel over the flat plate. For the T3C1 case,

**111**

**Figure 3.**

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications*

which represents the highest turbulence intensity test case among the T3C series test cases, the B-C model results are quite in agreement with the experimental data as the transition onset location is predicted with decent accuracy. For the T3C2 case, it is observed that although the B-C model predicted a good transition onset point, the turbulent stress abruptly rises after the onset. All other models predicted the

*Comparison of skin friction coefficients for the variable pressure gradient flat plate test cases.*

For the T3C3 case, it is observed that the γ-Reθ model [11], k-kL-ω model [14], and WA-γ model [24] outperform the other models as the B-C model prediction shows an early transition onset, whereas the one-equation γ model [22] predicts a rather late transition onset. For the T3C4 case, which represents the lowest Reynolds number case, all the models except for the B-C and WA-γ models show flow separation as their skin friction coefficients are below zero. Here, the B-C model obtained a quite good transition onset point that agreed with the experimental data although

transition onset location rather late in general.

*DOI: http://dx.doi.org/10.5772/intechopen.83520*

#### **Figure 2.**

*Comparison of skin friction coefficients for the zero pressure gradient flat plate test cases.*

*Transition Modeling for Low to High Speed Boundary Layer Flows with CFD Applications*

#### **Figure 3.**

*Boundary Layer Flows - Theory, Applications and Numerical Methods*

rise characteristic. Finally, for the T3A- case, the B-C [20], Menter et al. [22], Walters and Cokljat [14], and Nagapetyan and Agarwal [24] display early transition onset points with rather rapid rise in skin-friction, whereas two-equation Langtry and Menter [11] and Medida [31] models show quite good onset point and a gradual

**Case Uin Re<sup>∞</sup>** *Tu***%** S&K 50.1 3.4E+6 0.18 T3A 5.4 3.6E+5 3.00 T3B 9.4 6.3E+5 6.00 T3A- 19.8 1.4E+6 0.90 T3C1 5.9 3.9E+5 6.60 T3C2 5.0 3.3E+5 3.00 T3C3 3.7 2.5E+5 3.00 T3C4 1.2 8.0E+4 3.00 T3C5 8.4 5.6E+5 3.00

**Figure 3** depicts numerical and experimental skin friction coefficients for the T3C series variable pressure flat plate test cases. The T3C series flat plate test cases represent actual turbine characteristics by changing the pressure gradient by changing the upper wall profile of the wind tunnel over the flat plate. For the T3C1 case,

**110**

**Figure 2.**

rise in the skin friction.

*Inlet conditions for the flat plate test cases.*

**Table 1.**

*Comparison of skin friction coefficients for the zero pressure gradient flat plate test cases.*

*Comparison of skin friction coefficients for the variable pressure gradient flat plate test cases.*

which represents the highest turbulence intensity test case among the T3C series test cases, the B-C model results are quite in agreement with the experimental data as the transition onset location is predicted with decent accuracy. For the T3C2 case, it is observed that although the B-C model predicted a good transition onset point, the turbulent stress abruptly rises after the onset. All other models predicted the transition onset location rather late in general.

For the T3C3 case, it is observed that the γ-Reθ model [11], k-kL-ω model [14], and WA-γ model [24] outperform the other models as the B-C model prediction shows an early transition onset, whereas the one-equation γ model [22] predicts a rather late transition onset. For the T3C4 case, which represents the lowest Reynolds number case, all the models except for the B-C and WA-γ models show flow separation as their skin friction coefficients are below zero. Here, the B-C model obtained a quite good transition onset point that agreed with the experimental data although

the laminar region was rather inaccurate. Finally, for the T3C5 case, solution of the zero-equation B-C model [20], Menter et al. one-equation γ model [22], and WA-γ model [24] well agree with the experiment in the laminar region, the onset of transition is also fairly good with some delay, and again quite good agreement in the subsequent variable pressure gradient region is obtained.
