Procedure of inflow generation for turbulent boundary layer

1. Initialize the 3D flow field

- U: Spalding law (In the textbook 'Viscous Fluid Flow' of White, Eq. 6-41) is used for inner scale and the Cole's law of the wake (In the textbook 'Viscous Fluid Flow' of White, Eq. 6-47) is used for outer scale.

Here, the blending function (Eq. (16) in Lund et al.) is applied in the range of 0 < y < delta. Above y=delta, the Cole's law is employed only.

- V: Wall-normal velocity is introduced using the continuity relation.

- W: Impose zero velocity.

- Random flucutuations with a maximum amplitude of 50% of the free-stream value are superimposed on the mean values. For more than 50%, you should use Dt=Dt/20 for the first 20 time steps.

 

2. Calculate the mean and fluctuating velocities at the recycling position. The mean velocities are averaged using Eq. (18) of Lund et al. to exclude starting transient.

3. Calculate the u_tau (u_tau_rc) and theta (theta_rc) at the recycling position using the mean velocities.

4. Calculate the u_tau (u_tau_inlt) at the inlet position using Eq. (17) of Lund et al. which is similar to the Ludwig-Tillmann correlation. Here, the inlet momentum thickness (theta_inlt) is fixed.

5. Calculate the ratio of u_tau (Gamma) (See Eq. (6) of Lund et al.).

6. Scale the inner and outer mean velocities (U_inlt_inner, V_inlt_inner, U_inlt_outer, V_inlt_outer) at the inlet position using Eq. (4), (5), (9), (10), (15) and (16).

- Linear interpolation is used for scaling. W_inlt_inner and W_inlt_outer is not scaled.

- To scale U_inlt_outer, we should know delta_inlt in advance (see Eq. (5) of Lund et al.).

6.1 Guess the delta_inlt.

6.2 Calculate U_inlt_inner (Eq. (4)) and U_inlt_outer (Eq. (5)).

6.3 Blend U_inlt_inner and U_inlt_outer using the weighting function (Eq. (15) of Lund et al.). Now, we get U_inlt (Eq. (16)).

6.4 Calculate the momentum thickness at inlet (theta_inlt) using U_inlt. If this is not the same as the targeted theta_inlt. Go back to 6.1.

** Newton's method is employed to guess the (k+1)th delta_inlt.

- delta_inlt(k+1)=delta_inlt(k) - ( theta_inlt(delta_inlt(k))-theta_inlt(target) ) / (del theta_inlt / del delta_inlt)

- Here, del theta_inlt / del delta_inlt=( theta_inlt(delta_inlt(k)+epsilon) - theta_inlt(delta_inlt(k)-epsilon) ) / 2epsilon, where epsilon=10^(-10 )

- When | theta_inlt(target) - theta_inlt(k) | < 10^(-5), we finally get delta_inlt. Otherwise, you should repeat guessing the delta_inlt. The maximum iteration number is set to 100.

6.5 Like U_inlt, you can get V_inlt using Eq. (9), (10), (15) and (16).

7. Scale the inner and outer fluctuating velocities (u_inlt_inner, v_inlt_inner, w_inlt_inner, u_inlt_outer, v_inlt_outer, w_inlt_outer) using Eq. (13), (14), (15), (16).

7.1 Calculate u_inlt_inner (Eq. (13)) and u_inlt_outer (Eq. (14)).

7.2 Blend u_inlt_inner and u_inlt_outer using the weighting function (Eq. (15) of Lund et al.). Now, we get u_inlt (Eq. (16)).

7.3 Like u_inlt, you can get v_inlt and w_inlt.

8. Impose the inlet boundary condition (UBC1=U_inlt+u_inlt, UBC2=V_inlt+v_inlt, UBC3=w_inlt).

9. Impose the far-field boundary condition (VBC) using linear regression.

9.1 Calculate the displacement thickness using time-averaged flow field (see Eq. (18) in Lund et al.) at each station.

9.2 Calculate the v at far-field using linear regression. Please see the 'Estimating beta(the slope)' in http://en.wikipedia.org/wiki/Linear_regression .