Numerical experiment procedure

The present study focuses on the steady and unsteady flow field in the transonic wind tunnel. The computations are done by systematically changing wind tunnel configuration just like done in the experiments. The computational domain includes all the components shown in Figure 1, and is treated by two-dimensional manner. Any effects of sidewall boundary layer are not modelled in the present study. Total pressure, total temperature, and the flow angle are fixed at the inlet while atmospheric static pressure is assumed at the outlet. The suction effect on the transonic wall is modelled by specifying constant wall-normal velocity on the wall.

The present steady simulation focuses on the effects of the transonic wall suction, throttle opening angle, and tailboard angle on the pitchwise periodicity of the flow field. For unsteady computations, the experimental influence coefficient method 06 is simulated by oscillating only the centre blade (Blade No. 0) along normal to the chord line.

Evaluation of steady and unsteady flow fields

The steady flow field upstream and downstream the cascade is evaluated by Mach number M, flow angle β, and static pressure p. In addition to these distributions, the periodicity downstream the cascade is assessed by the following two simple indicators:

The Δp/p_t quantifies the pressure difference between two centre channels. The TV is an indicator named “total variation,” which quantifies global periodicity by the sum of pressure difference between two neighbouring channels.

The steady aerodynamic force acting on the blade is evaluated by pressure and lift coefficients defined as follows.

The reference static pressure is obtained at 50%c upstream from the leading edge (i.e., ξ₁=0). On the other hand, unsteady static pressure and aerodynamic force under the blade oscillation h(t)=h¯eiωt are evaluated by the Fourier transformation of the C_p and C_l as follows.

Flow solver

The baseline code employs finite-volume discretization of compressible Reynolds-averaged Navier-Stokes equations on the structured mesh which have been developed in our past study 14. Inviscid and viscous terms are evaluated by the AUSM-type SHUS scheme 12 with the third order MUSCL interpolation and the second order central difference, respectively. Time integration is conducted by the first order backward Euler scheme for the steady flow analysis, while the three-point backward difference with three Newton iterations is used for the unsteady simulations. As a turbulence model, one-equation Spalart-Allmaras model 13 is employed with fully-turbulent treatment.

CFD mesh and overset mesh technique

Our key progress for enabling simulations of whole the wind tunnel is the extensive use of the overset mesh technique. The every component of the wind tunnel is meshed by simple O- or H-mesh with algebraically extruding the surfaces, and they are assembled like a patchwork quilt. The data communication is conducted by the trilinear interpolation.

The connectivity data is generated by an algorism that combines distance-based blanking and iterative adjustment of the fringe cells 04. Figure 2 shows a process of creating the connectivity data. Before searching data source (donor) for interpolation, the cells are blanked based on the distance of the components. Then the fringes of the activated cells are marked as receptor, and donors are searched. The cells that do not have donors are considered as “orphan” cells. After finishing the donor search, the blanked cells neighbouring the orphans are activated and donor search is run again. The dataset is completed after all orphan cells are eliminated. In the present computation, the receptor cells cannot be donors, and any orphan cells are not allowed since they bring significant numerical error 10.

Figure 2.

Iterative donor search process. Red cells: orphan, blue cells: non-orphan. (a) Initial blanking, (b) After 2 iterations, (c) After 4 iterations, (d) Final mesh.

Figure 3 shows the mesh assembly for the present study. The mesh system has two cells in the spanwise direction and 0.75 million cells in total. All the inter-blade passages, bypass channels, and small gaps are successfully connected around the midpoint between the parts. The percentage of activated cells against the total cells is 72.4%.

Figure 3.

Overset mesh system of the wind tunnel. Only active cells are shown here.

In addition to the wind tunnel mesh, a conventional line-matched multi-block mesh is generated for preparing baseline flow field with perfect pitchwise periodicity. This configuration is referred to as “ideal” case. Figure 4 and Table 2 shows the appearance and mesh parameters for the O-mesh around the blade. The multi-block configuration have slightly higher mesh resolution.

Figure 4.

Mesh around the blade 0. (a) Wind tunnel, (b) Multi-block.

Baseline flow field

At the beginning of the discussion part, the baseline flow field for the discussion hereafter is presented for both wind tunnel and ideal configurations. The back pressure for the both cases are 0.67p_t. The wind tunnel settings are: θ_ts = 6, θ_tb = 2.4, θ_th = 1.2 degrees, and the suction velocity of v_s = 6 m/s.

Figure 5 shows Mach number distributions for the both cases. Figure 5 and Figure 5 show the wind tunnel case. The supersonic started inflow is realized for all passages of the cascade and the bypass channel. In the blade passage, the oblique shock from the leading edge and normal passage shock can be observed. The shock pattern including lambda-shaped shock-boundary layer interaction within the passage for the wind tunnel case is quite similar with that in the ideal case shown in Figure 5.

Figure 5.

Mach number distributions for the baseline case. (a) Whole flowfield through the nozzle throat, test section, and downstream the cascade, (b) Wind tunnel configuration, (c) Ideal, infinite cascade.

Figure 6 shows C_p distributions on the blade No. −2 to 2 and the ideal case. As expected from the Mach distributions, almost periodic flow field is realized for the wind tunnel case. The positions of the shock are deviated by approximately 5%c within the blade −1 to 2 both on the suction and pressure sides.

Figure 6.

Cp distributions for the baseline case.

Effect of suction on the upstream periodicity

The effect of transonic wall suction is evaluated for making the inflow as uniform as possible. Figure 7 and Figure 7 show comparison of shock pattern upstream the cascade, which is visualized by the horizontal density gradient. The lower bypass is unstarted and significant non-uniformity can be seen in the non-suction case, while appropriate amount of suction gives uniform shock structure as shown in Figure 7.

Figure 7.

Effect of transonic wall suction on the inflow uniformity. (a) No suction, (b) v_s = 6 [m/s], (c) Mach number and flow angle along ML1.

Figure 7 shows detailed comparison of Mach number and flow angle along ML1 against different suction velocities. Both Mach number and flow angle are significantly affected by the amount of suction, and the effect of suction mainly appears upstream the blade 2 and 3. In addition, an appropriate amount of suction gives similar distribution as the ideal case.

Effect of throttling on the downstream periodicity

The effect of throttling angle on the downstream periodicity is evaluated by fixing all other parameters. Figure 8 shows a comparison of static pressure distributions along ML2 for five different throttling opening angles: θ_th = 0.4, 0.8, 1.0, 1.2, and 1.6[deg]. Under the fixed tailboard, the pressure distribution downstream the cascade is highly affected by the throttling angle. For the θ_th = 0.4 [deg] case, the pressure downstream the blade No. −3 to −1 is significantly small compared to that downstream the blade No. 1 to 3. The situation is opposite for the θ_th = 1.6 [deg] case.

Figure 8.

Effect of throttle closing angle on the wall pressure distribution along ML2.

The blade loading is also highly affected by this non-periodic pressure distribution. Figure 9 shows the C_p distributions among five centre blades for the θ_th = 0.4 and θ_th = 1.6 [deg] cases. There are significant variations in the blade loading and shock position for these cases, compared to the θ_th = 1.2 [deg] case shown in Figure 6.

Figure 9.

Cp distributions with non-uniform back pressure.

This result is qualitatively consistent with the findings by Lepicovsky et al. 08, and it also implies that there is an optimal throttling angle for realizing the best periodicity for every different tailboard angle.

In order to confirm the existence of optimal throttling angle for the best periodicity, the periodicity in static pressure is quantified by parameters Δp/p_t and TV in Equations (–). The optimum conditions for these indicators are Δp/p_t = 0 and the minima of TV. Figure 10 shows the change in Δp/p_t and TV for different tailboard and throttling angles. Each lines are obtained by sweeping θ_th while keeping θ_tb constant. The zero-points of the Δp/p_t and the local minima of TV can be found on each tailboard angles. Therefore, it can be confirmed that the throttling angle for the best periodicity depends on the tailboard angle.

Figure 10.

Effect of tailboard and throttling angles on the non-periodicity indicators.

Controlling pitchwise periodicity and its importance

In the choked cascade with fixed supersonic inflow, all the aerodynamic characteristics are dependent variables of the outlet pressure. Here, the effect of periodicity on the measured aerodynamic characteristics is discussed in detail.

Figure 11 shows the relationships between spatial averaged outlet pressure p2¯/pt and outflow angle β2¯. The averaging is conducted over two centre passages along ML2 (i.e., −s≤ξ2≤s). In the wind tunnel results, the measured p2¯/pt−β2¯ characteristics under the fixed tailboards depend on the tailboard angle, and none of them is consistent with the cascade under ideal periodicity. On the other hand, although the difference of approximately 0.2 degrees exists, the p2¯/pt−β2¯ curve under Δp/p_t = 0 shows similar trend with the ideal case.

Figure 11.

Effect of downstream uniformity on the outflow characteristic of the cascade.

From this result, it can be concluded that realizing the best pitchwise periodicity is quite important for obtaining appropriate aerodynamic characteristics of ideal cascades.

Aerodynamic influence coefficients

Figure 12 shows the comparison of frequency dependency of the AICs on the three centre blades between wind tunnel and ideal configurations. Since the AICs are expressed by complex number, their amplitude and phase angle are shown. The trend of two different configurations are qualitatively similar both in amplitude and phase angle. However, significant difference in amplitude can be seen in the frequency ranging from 400 Hz to 800 Hz, especially for the blade −1.

Figure 12.

Comparisons of frequency dependency of AICs between ideal and wind tunnel configurations. (a) Amplitude, (b) Phase difference against blade displacement.

Blockage effect of the tailboards

In order to find out where the difference in the amplitude of AICs originates, unsteady flow field around the cascade is investigated. Figure 13 shows pressure amplitude around the cascade for both ideal and wind tunnel configurations. Four frequency levels from 250 Hz to 700 Hz are highlighted in order to see the increasing process of the difference.

Figure 13.

Pressure amplitude around the oscillating blade. (a) Ideal, infinite cascade. White line shows tailboard in the WT setup, (b) Wind tunnel configuration.

In the ideal cascade shown in Figure 13, pressure amplitude continuously increases at downstream of the blade −2 to −1 as the frequency increases. In other words, the pressure amplitude becomes higher on the downstream of lower blades.

In the wind tunnel configuration shown in Figure 13, the tailboard exists where the pressure amplitude increases in the ideal configuration. Therefore, there can be some interaction between the wind tunnel wall and acoustic wave propagating toward minus-direction of the blade number. The acoustical reflection on the tunnel wall potentially cause the observed amplitude difference. More detailed interaction phenomenon and the consistency with the theoretical backgrounds could not be figured out however, it can be said that the unsteady flow field can be grasped in a qualitative sense by the developed CFD code.

Acoustic cut-on in the duct

Although the actual frequency is very high compared to the past flutter experiments (e.g., up to 550 Hz and 730 Hz for the standard configurations five and seven, respectively [05]), the acoustic cut-on condition in the exit duct appears in the present computation. The lowest cut-on frequency of the acoustic wave propagating upstream and downstream inside a rectangular duct can be derived as follows.

In the present baseline case, the duct width, sound speed and Mach number are D=87.6 mm, α= 327 m/s, and M = 0.693, respectively. The resultant cut-on frequency is fcuton=1350 Hz.

Figure 14 shows the pressure amplitude inside the duct. Figure 14 and Figure 14 correspond to cut-off (1100 Hz) and cut-on (1500 Hz) case, respectively. The pressure amplitude decays toward downstream for the cut-off case, while the wall pressure becomes higher entire in the streamwise direction for the cut-on case. Figure 14 shows the amplitude of wall pressure against blade frequency sampled near the leading edge of the throttle. The pressure amplitude gradually increases as the frequency increases. Then, the amplitude drastically jumps when the blade frequency reaches the cut-on frequency. From this result, it is confirmed that the present method can capture global unsteady phenomena like acoustic cut-on.

Figure 14.

Unsteady pressure amplitude in the duct downstream the cascade. (a) 1100 Hz, cut-off, (b) 1500 Hz, cut-on, (c) pressure amplitude at the leading edge of the throttle plate.