## Abstract

A new class of power generation devices that experiences increased losses due to bulk flow separation in segments of their expected in-flight regime is emerging. As such, active flow control becomes increasingly relevant to mitigate these losses and reclaim the entire flight envelope. This study explores the effect of flow injection on transonic flows experiencing bulk separation. Reynolds-averaged Navier–Stokes simulations of a 3D wall-mounted hump at low Reynolds numbers are conducted to assess the response of transonic bulk separation to flow injection. Unsteady simulations are performed to understand the differences between slot and discrete port injection and determine optimum forcing frequencies. Discrete ports require higher pressures to overcome the momentum deficit associated with the smaller injection area relative to the width of the domain. Steady and unsteady injections are found viable strategies for mitigating the extent (or appearance) of bulk separation. Experiments are conducted with discrete injection for a range of Mach and Reynolds numbers. The response of the bulk separation to said injection is evaluated by analyzing both local pressure measurements and schlieren imaging. The study shows that the required pressure of injection is strongly correlated to the length scale of the uncontrolled separation. With large-eddy simulations, the flow separation and frequency content within the separated region can be reasonably predicted. This study aims to take further steps to establish guidelines for applying flow control to the emerging class of power generation devices experiencing losses from bulk separation.

## Introduction

Future thermal-based cycles are being developed with much higher loading in the low-pressure turbine (LPT) than state-of-the-art aero-engines [1]. This new generation of LPT is prone to bulk flow separation in low Reynolds conditions resulting in decreased performance and a loss in the available operating regime and mitigation of performance benefits. Passive and active flow control strategies to extend the performance of aerothermal devices have been of significant interest in recent decades. Passive flow control techniques that mitigate separation are accompanied by penalties at the design condition, making them less palatable [2]. Active flow control techniques potentially allow for a minimal impact on the design operating condition while providing considerable potential performance benefits when actuated in conjunction with the appearance of bulk separation.

Separated flows are often studied in flat plates with a contoured opposite wall to replicate the pressure gradient in low-speed testing [3] or using wall-mounted humps to consider the effect of the local curvature [4]. Such geometries allow for a wide range of conditions to investigate the coupling of the bulk separation with Mach and Reynolds numbers. Earlier studies involve using wall-mounted humps to characterize separation and observe potential performance benefits from the addition of flow control. Luedke et al. [5] studied the effects of both slot and discrete, steady flow injection. Saavedra and Paniagua show the impacts of transient inlet conditions experimentally, particularly in rapid start-up [4] and sinusoidal injection numerically [6] on wall-mounted humps. A similar study with a slightly elevated inlet Mach number was conducted by Seifert and Pack [7], where periodic suction and blowing were applied. These works show that steady and unsteady injections and suctions can mitigate boundary layer separation in subsonic internal flows. Other studies investigating alternative active control methods, such as dielectric discharge, were conducted by Pescini et al. [8] and Martinez et al. [9].

Flow control with plasma actuators has been demonstrated in several applications, including critical airfoils [10]. Greenblatt and Wygnanski [11] documented the abatement of flow separation with periodic hydrodynamic excitation. The present study elevates inlet Mach numbers to force the crest of the wall-mounted hump to reach transonic conditions and focuses on applying steady and unsteady flow injections to separating flow in a low Reynolds regime. The flow injection site of the wall-mounted hump is near enough to the summit of the hump to be considered transonic. Computational studies of increasing fidelity were conducted to characterize the uncontrolled separation and the introduction of flow injection. Blow-down experiments were conducted to compare the findings of the numerical studies.

At this point, it is worth noting that this effort is part of a larger initiative to investigate the feasibility of incorporating active flow control technologies in a high lift/high work low-pressure turbine stage. References [2,49] are quite representative of the available literature on flow control technologies in that although many techniques have shown considerable promise in the laboratory over the years, there is very little regarding applications of flow control in the more complex flow fields associated with operating turbine stages. The approach is to investigate the feasibility of applying flow control to a turbine stage by conducting a series of experiments with geometries of increasing complexity, as laid out by technology readiness levels [12], and ultimately arriving at a demonstration of flow control in a transonic, rotating turbine. In this first step toward increased fidelity of experiments, the focus is on the effectiveness of flow control at transonic conditions. It is very little in the way of the open literature on this topic save for studies concerned with external flows.

A family of transonic turbines was designed to the same work (2.80) and flow coefficients (0.78) as the stage described in Ref. [13], albeit with different levels of the Zweifel coefficient (Zw) in the blade row and with a substantially higher design target efficiency. These turbine stages are designated LXFHW-HS where the acronym is as follows: L indicates an LPT blade, X is a wildcard that represents the mean-line loading level of the blade (e.g., 1: Zw = 1.32, 2: Zw = 1.60, and 3: Zw = 1.78). Each airfoil is front-loaded with an exit Mach number of order 0.8, and it is part of a high work stage (HS).

The loading distributions over the LXFHW-HS family of blades at midspan were predicted via a two-equation Reynolds-averaged Navier–Stokes (RANS) solver. The results are shown in Fig. 1 in terms of the pressure coefficient versus the fraction of the axial chord. The midspan airfoil geometries are also displayed in the figure. All three airfoils are front-loaded, and the suction side pressure distributions over the airfoils are characterized by various levels of acceleration and deceleration in the vicinity of peak suction. The transonic hump studied in this experimental campaign is consistent with the loading characteristics over the LXFHW-HS embodiments of the airfoils. It explores a more extreme set of pressure gradients in the vicinity of peak suction than that occurring on the turbine passages. In this study, the demonstration of flow control for the transonic hump justifies further exploitation in high-speed annular turbine cascades.

Fig. 1
Fig. 1
Close modal

## Research Methodology

The objective of the present study is to characterize the behavior of uncontrolled bulk separation, devise a relatively unobtrusive flow control device, and implement strategies that eliminate bulk flow separation. To meet these objectives, the 3D geometry from Saavedra and Paniagua [4,6] was used. No changes were made to the geometry other than the addition of the flow control port, which is described in Sec. 4.

### Numerical Methodology.

The numerical domain used for the computational fluid dynamics (CFD) simulations is presented in Fig. 2(a). There, a slice of the actual 3D domain is shown. The test article is placed in the middle (marked as a continuous line), and upstream and downstream buffer length have been added to the numerical domain to place the hump far from the boundary conditions. The inlet length has been chosen as the one that ensures a turbulent boundary layer in a flat plate approximation, while the outlet is placed more than three hump chords downstream.

Fig. 2
Fig. 2
Close modal

The x-axis denotes the streamwise dimension, y-axis is the vertical one, and z-axis is spanwise. The y-coordinate of the top wall of the domain was determined by measurements taken from the physical test article.

It was decided to include the flow injection ports in the numerical domain as the boundary conditions for the flow control are measured upstream of the elbow (shown in Fig. 2(b) inset). This introduced an unknown but not negligible pressure loss and complex flow features that are analyzed in Sec. 5. Figure 2(b) includes all the injection ports.

The first RANS simulations are done in 2D, modeling the ports as a continuous slot. For the 3D RANS simulation, the width of the domain is the width of one port and the space between them. This is the space left between the two vertical plates in Fig. 2(b). Alongside periodic boundary conditions on the sides, this reduction in the computational domain ensures the same accuracy as the whole test article. This can be done as Saavedra and Paniagua found the central 10% span of the test article showed the same pressure distribution for subsonic flows [6]. Due to the grid requirements, large-eddy simulation (LES) simulations do not include the port for the base flows.

The mesh employed for the 2D RANS cases was created using ICEM CFD and is shown in Fig. 3(a). It is a structured mesh with 347,481 elements, y+ = 1 on the walls, and a minimum quality of 0.82. Polyhedral elements are used for the 3D mesh, as shown in Fig. 3(b), with a total element count of 581,372, done in ansys fluent. The lowest element quality is 0.09 and the y+ is 1 on the wall surface and 4 on the duct surface.

Fig. 3
Fig. 3
Close modal

As one of the main concerns of this study is the prediction of the separation and reattachment points, the transition SST turbulence model is used. Compared to results with the kω SST model, the latter overpredicts the length of the separation bubble (SB), although both predict the same separation point.

The 2D RANS simulations were solved with the steady (for cases without pulsation) pressure-based solver, while the 3D RANS was solved employing the unsteady density-based solver. The discretization scheme was the second-order upwind for space and second-order implicit for time, with a step size of 0.01 ms. The 2D simulations to assess pulsated injection used a time-step equal to the period of pulsation divided by a factor of 1000.

The boundary conditions are the following:

• Total pressure and temperature imposed at the inlet of the domain and the blowing.

• Symmetry wall on the top surface.

• Viscous and isothermal wall on the bottom surface.

• Static pressure is imposed at the outlet, with non-reflective boundary conditions.

• Periodic boundary conditions on both sides.

It is important to note that the wall is assumed to be isothermal since the duration of the experimental blow-down is short enough to neglect the increase in the temperature due to the heated fluid.

The ideal gas equation, the Sutherland equation for viscosity, and the temperature dependent thermal conductivity have been used.

A convergence study for the 2D case was performed using a Reynolds number of 2.5 × 106 m−1 and a Mach equal to 0.5 at the inlet using the method laid out by Celik et al. [14]. Table 1 is the result of the study using the separation bubble length, total pressure loss, viscous drag on the bottom wall, and the area-weighted mass flow from the meshes investigated.

Table 1

Grid convergence study of 2D mesh

Mesh IDNodesrMean eaMean GCI
Coarse2199961.2511.27%0.80%
Mid347564
Fine4888311.190.62%0.11%
Mesh IDNodesrMean eaMean GCI
Coarse2199961.2511.27%0.80%
Mid347564
Fine4888311.190.62%0.11%

After the convergence study, the medium mesh was chosen as the extrapolated error between the medium and fine mesh was small. The required computational time for the medium mesh was significantly less than the computational time needed for the fine mesh.

The numerical approach used to conduct LES is slightly different. For the base case, the ports are removed on the geometry, which is justified by several numerical tests conducted with the RANS approach.

The domain size has been modified to reduce the grid requirements. The downstream length was reduced from 463 mm to 163 mm, while the upstream length was modified from 200 mm to 180 mm (see Fig. 2(a)).

The mesh, shown in Fig. 3(c), is fully structured and uniformly extruded in spanwise direction. The width has been increased from 1 port to 7 ports, equivalent to 0.2 chords of the hump (as done by Franck and Colonius in their LES of a wall-mounted hump [15]), which will aid in capturing larger turbulent structures. The domain has meshed in ICEM CFD. The total number of nodes is 76 million, while the lowest element quality is 0.67. The mesh parameters for LES are presented in Table 2.

Table 2

Mesh parameters used for the LES case

DimensionMin +Max +Mean +Nodes
X38250781070
Y0.525048320
Z202020222
DimensionMin +Max +Mean +Nodes
X38250781070
Y0.525048320
Z202020222

The spatial discretization is a central difference scheme, and a second-order temporal discretization is used. The subgrid-scale model used is the dynamic Smagorinsky–Lilly, following the recommendation in Ref. [15]. While the boundary conditions are equivalent to RANS, there are some noticeable differences:

• Inlet turbulent perturbations using the vortex method generator are included.

• Profiles of total pressure and temperature, extracted from the RANS, are set at the inlet. This helps to reduce the inlet length of the LES domain while keeping the same accuracy.

• Similarly, profiles of turbulent kinetic energy and energy dissipation rate, based on the RANS results, are imposed.

Moreover, the time-step has been reduced to 7 μs, ensuring a maximum Courant–Friedrichs–Lewy number of 39.8 and an average of 2.3.

To obtain steady statistics, the average time is 10 ms, which corresponds to two times the expected residence time of a particle in the domain. The simulations were initialized with the RANS solution.

### Experimental Methodology.

The wall-mounted hump was designed to fit into the linear test section of the Purdue Experimental Turbine Aerothermal Laboratory (PETAL) wind tunnel described and sketched in Paniagua et al. [16]. The contraction nozzle from a circular to a square cross section was carefully optimized and evaluated to deliver uniform inlet conditions to the test article [17]. The performance of the test article, shown in Fig. 4, is assessed through wall static pressure taps, total pressure and temperature probes, and schlieren imaging. The location of the total pressure and temperature probes can be seen in Fig. 2(a), where (p) denotes a Kiel–Pitot probe and (T) denotes a total temperature probe. The distribution of wall pressure taps is shown in Fig. 4. Pressure measurements are acquired with ScaniValve miniature pressure scanner units (50 psid). Temperature measurements are recorded with VTI thermocouple acquisition devices and k-type thermocouple junctions. A linear schlieren setup is used to analyze uncontrolled flow separation and steady injection flows.

Fig. 4
Fig. 4
Close modal

Inlet Mach number is calculated from a pressure tap on the top wall of the wind tunnel and a Kiel-Pitot probe. These taps are placed 35 mm from the start of the test article. The Mach number combined with total temperature and Sutherland’s law for viscosity yielded the experimental Reynolds number. No reference length has been chosen for the Reynolds so that the application of the findings can be applied to other studies where the geometry may be unlisted. As a result, Reynolds per meter is reported. Schlieren imaging was taken at the crest of the wall-mounted hump to study the formation of the separation at 10 kHz. While the behavior of the full blowdown and the development of flow and its interaction with the ports is studied using 2 kHz frame rate. The location viewed with schlieren can be seen in Fig. 2(a), while the setup is shown in Fig. 4.

The preliminary numerical study boundary conditions are targeted for the experimental set points to observe separated flows. These points are set via mass flow, temperature, and vacuum tank pressure. As the test section is not isolated from the vacuum tank by a sonic orifice, the back pressure of the test section varies. Similarly, the Reynolds and Mach numbers at the inlet evolve during any blowdown. Each operating point was taken as a 0.20 s interval during a 10 s blowdown where quantities are averaged to characterize a specific Mach and Reynolds pair. These pairs are targeted to investigate the behavior of bulk separation and its response to flow control, with small variations in total temperature caused by different flows through the tunnel before the set point.

The different experimental and numerical cases used in this paper are collected in Table 3. The cases whose name is preceded by an “H” are tested with hot air, while an “N” indicates cold nitrogen. Details regarding the flow control added are presented in the table and described in Sec. 5.

Table 3

List of cases/experiments

NameRe (m−1 ×106)Mach (–)PR (–)f (Hz)
H-01-1962.60.600
H-01-3902.50.575
H-01-5882.40.550
H-01-7492.40.530
H-01-8382.40.520
H-02-8552.80.530
H-02-8882.80.524
H-03-7923.20.530
H-04-3813.90.560
H-04-5483.80.575
H-04-7083.80.550
H-04-8723.70.530
H-04-9383.70.520
H-05-0804.30.575
H-09-5423.80.5753.920
H-11-5413.80.5752.920
H-14-5912.40.5753.660
H-17-7533.70.5752.010
H-20-3743.90.5751.400
N-15-7723.80.5752.70200
N-17-7903.80.575
N-19-6153.50.5753.4750
N-23-6263.60.5753.42125
N-24-6483.60.5753.48138
N-22-5913.60.5753.38150
N-20-6813.70.5753.22200
N-21-7893.80.5753.20400
NameRe (m−1 ×106)Mach (–)PR (–)f (Hz)
H-01-1962.60.600
H-01-3902.50.575
H-01-5882.40.550
H-01-7492.40.530
H-01-8382.40.520
H-02-8552.80.530
H-02-8882.80.524
H-03-7923.20.530
H-04-3813.90.560
H-04-5483.80.575
H-04-7083.80.550
H-04-8723.70.530
H-04-9383.70.520
H-05-0804.30.575
H-09-5423.80.5753.920
H-11-5413.80.5752.920
H-14-5912.40.5753.660
H-17-7533.70.5752.010
H-20-3743.90.5751.400
N-15-7723.80.5752.70200
N-17-7903.80.575
N-19-6153.50.5753.4750
N-23-6263.60.5753.42125
N-24-6483.60.5753.48138
N-22-5913.60.5753.38150
N-20-6813.70.5753.22200
N-21-7893.80.5753.20400

## Base Flow

The 2D representation of the centerline of the test article is used to build the target conditions of the experimental campaign. Once the experiment was conducted, the numerical domain is altered to match the experimental throat height, and the measured boundary conditions at instances of interest are run for direct comparison.

The definition of pressure coefficient (Cp) employed is
$Cp(p)=Pt,in−PPt,in−Pout$
The definition of friction coefficient (Cf) employed is
$Cf=τw12ρV2$

### Low Frequency Wall Measurements.

Figure 5(a) shows the emergence of Reynolds lapse behavior without flow control. The separation bubble is represented by the sudden change in the slope of the pressure coefficient, just after the crest. This change in slope is more horizontal and extends farther the stronger the SB.

Fig. 5
Fig. 5
Close modal

Another behavior becomes apparent as inlet Mach number is analyzed for a constant Reynolds number, shown in Fig. 5(b). The SB tends to re-attach closer to the crest for lower inlet Mach. As the passage, overall, becomes higher speed, the flow is less able to adhere to the curvature and experiences a more dominant bulk separation.

In Fig. 6, the Cp distribution from numerical studies is compared with those from the experiments. The onset of the separation is predicted in the same location by the RANS and LES, just before the crest, where Cf becomes negative and Cp suddenly drops. Then, there is the separation bubble with the pressure plateau and a final reattachment where the RANS is slightly shorter than the LES. Note that, where the separation ends, the pressure coefficient is already decreasing, tending to be asymptotic. As opposed to the numerical approach, the experiment exhibited the onset slightly after the crest, as marked by the flat region in Cp. The reattachment occurs at around x/L = 0.5 experimentally, while for both simulations reattachment occurs near 0.8.

Fig. 6
Fig. 6
Close modal

The pressure gradient varies from strongly favorable to strongly adverse near the LXFHW-HS airfoils peak velocity along the suction side. The same is observed for the transonic hump. Accordingly, the pressure coefficient at the suction peak, the Mach number at that location, and the fraction of surface distance where minimum pressure occurs describe the pressure variation over each geometry. Additionally, one can take the fraction of surface distance between the re-laminarization limit $K=νU2dUds=3.0×10−6$ and suction peak as a measure of the strength of the favorable pressure gradient upstream of the point of minimum pressure.

Similarly, the fraction of surface distance between the point of minimum pressure and the value of Thwaites’ pressure gradient parameter associated with separation $λθ=KReθ2=−0.09$ can be obtained as a measure of the strength of the adverse pressure gradient downstream of peak suction. These values are given in Table 4 for the LXFHW-HS airfoils and the transonic humps in this study. One can readily see that the transonic humps investigated here exhibit a more extreme level of loading than the LXFHW-HS series. Therefore, they are a good first test for the applicability of flow control to high lift, high work LPT blades.

Table 4

Pressure gradient characteristics of the LXFHW-HS airfoils and the transonic humps

GeometryMmaxCp,maxs/stot @ Cp,maxs/stot @ Krelams/stot @ Krelam, Cp,maxs/stot @ $KReθ,sep2$s/stot @ $KReθ,sep2−s/stot$ @ Cp,max
L1FHW-HS0.991.360.2520.226−0.0260.4350.183
L2FHW-HS1.111.610.2820.223−0.0590.4040.122
L3FHW-HS1.372.020.1910.172−0.0190.3700.179
H-01-8381.112.890.2610.260−0.0020.2630.002
H-04-9381.142.630.2610.258−0.0030.2630.002
GeometryMmaxCp,maxs/stot @ Cp,maxs/stot @ Krelams/stot @ Krelam, Cp,maxs/stot @ $KReθ,sep2$s/stot @ $KReθ,sep2−s/stot$ @ Cp,max
L1FHW-HS0.991.360.2520.226−0.0260.4350.183
L2FHW-HS1.111.610.2820.223−0.0590.4040.122
L3FHW-HS1.372.020.1910.172−0.0190.3700.179
H-01-8381.112.890.2610.260−0.0020.2630.002
H-04-9381.142.630.2610.258−0.0030.2630.002

### Flow Topology.

A comparison between the mean flow in the RANS and LES results is shown in Fig. 7. The Reynolds per meter is 2.5 million, and the inlet Mach number is 0.575, corresponding to the case H-01-390 in Table 3.

Fig. 7
Fig. 7
Close modal

Both RANS and LES results exhibit a massive separation. In the RANS results, a small normal shock wave appears on the suction peak, due to peak Mach numbers near 1.2. This does not happen for the time-averaged LES results due to the lower maximum Mach number below 1. The reason is the lower pressure losses predicted by the RANS simulation, which allow an increase in the Mach number.

Figure 8(a) shows a certain time-step for the mid-plane contours. It is appreciated that smaller structures appear on the separated region. Also, several waves appear on the high-speed region above the hump. These waves propagate upstream, resulting from the channel effective height variation due to the separated shear layer blockage. Those are also seen in the schlieren image in Fig. 8(b), showing a similar position for the onset of separation.

Fig. 8
Fig. 8
Close modal

The structures that occur on the wall surface are analyzed in Fig. 8(c), where the wall shear stress and heat flux for a certain time-step are shown with the qualitative shape of the hump for reference. The shear stress reaches its peak just after the crest, where the onset of separation is located. Then, it drops to negative values just below zero as a result of the separation.

Note that the contours of wall shear stress and heat flux show similar trends. These magnitudes are representative of the derivative of a magnitude with respect to the normal wall direction, whether it is the tangential speed or the temperature. This means that both fields are equivalent if the Prandtl number is near 1 (0.71 for this case). Near the inlet of the computational domain, Fig. 8(c) shows long coherent streaks of turbulent structures. At the base of the hump, the structures become shorter and more chaotic, and eventually disappear while the flow is being accelerated: this is indicative of the re-laminarization of the boundary layer. Downstream of the inception of the separated region, the characteristic size of the turbulent structures is considerably smaller, indicating an increase in the turbulence of the flow.

Figure 9 provides a frequency analysis of the pressure field from the LES, which was sampled at 36 kHz according to the time-step size. Wall static pressure was experimentally monitored using pneumatic lines connected to low frequency pressure transducers preventing identification of frequencies above 50 Hz. Different frequency contents are noticed at each probe:

• x/L = 0.04. This probe is only affected by the inlet turbulent perturbations. The harmonics of 250 Hz, 1 kHz, and 2 kHz are shown, while for 500 Hz the frequency content is significantly attenuated.

• x/L = 0.15. The re-laminarization of the boundary layer occurs here, as a result, the frequency content is more uniform.

• x/L = 0.22. Just after the separation, 200 Hz, 400 Hz, and 900 Hz appear as main peaks.

• x/L = 0.45. In the main extent of separation 400 Hz is the dominant frequency associated with shedding.

• x/L = 0.75 Just upstream of reattachment, the 900 Hz content gets damped as the high frequency shedding is dissipated in the domain, while a 200 Hz peak is present.

• x/L = 1.21. After reattachment, the fast Fourier transform (FFT) shows a pink noise pattern where the content decreases asymptotically with the frequency and no dominant peaks occur. The flow is already attached and mixed.

Fig. 9
Fig. 9
Close modal

The main shedding at 400 Hz, which corresponds to a Strouhal number of St = 0.0793 using the hump height (36 mm) as reference length, can be seen with the unsteady pressure contours on the mean plane in Fig. 10. These large structures, that can be noticed as low-pressure regions, propagate downstream while created at such frequency.

Fig. 10
Fig. 10
Close modal

Faster vortical structures can also explain the local 900 Hz peak, but they do not separate from the suction peak before being dissipated.

Figure 11 highlights the Mach contours near the inception of separation for different time-steps to see the behavior of the shock wave. As discussed in Fig. 7(b), no normal shock wave is predicted with mean flow, but shocks can appear in the transient behavior of the flow. At certain instances in time, a supersonic region can appear. Then, in the order of 1 ms, the shock created due to the blockage of the separation, moves upstream while losing speed and disappearing. Focusing on the schlieren images, an equivalent propagation of the waves in the upstream direction is appreciated. During this process, the inception of the bubble is barely affected.

Fig. 11
Fig. 11
Close modal

## Flow Control Design

The optimal location for the flow control implementation is found based on the global stability analysis [18,19]. The fully compressible non-linear Navier–Stokes equations can be written in the matrix form
$∂∂tq=L(q)q$
(1)
where q represents the flow quantities q = [ρ, u, v, w, T] which can be decomposed into the mean (or equilibrium) $q¯$, and fluctuating q′ components, i.e., $q=q¯+εq′$. Substituting this equation in Eq. (1) and retaining terms that are larger than $O(ε2)$ gives
$∂∂tq¯+∂∂t(εq′)=L(q¯)q¯+L(εq′)q¯+εL(q¯)q$
(2)
Considering that Eq. (1) also holds for the mean quantities, Eq. (2) can be simplified to $∂∂tq′=L(q′)q¯+L(q¯)q′$, or in the compact form as
$∂∂tq′=N(q¯)q′$
(3)
where $N(q¯)$ is a function of spatial derivative operators and mean flow quantities, independent of time. Modal decomposition of perturbations follows:
$q′=q^n(x,y,z)e−jωnt$
(4)
where for the mode n, $q^n$ represents the mode shape and ωn indicates the complex phase speed. A mode’s frequency and growth rate are determined by the real and imaginary parts of ωn, respectively. In this study, we assume that the flow and perturbations are periodic in the spanwise direction, z, and therefore
$q′=q^n(x,y)e−j(ωnt−βz)$
(5)
where β is the spanwise wavenumber. Substituting Eq. (5) into Eq. (3), gives a generalized eigenvalue problem
$Aq^n=ωnBq^n$
(6)
where A and B matrices can be found in Ref. [20]. The base flow quantities are calculated using ansys fluent at the inlet Mach number of 0.1 and Reynolds number per-unit length of 2.17 × 106m−1.

The linearized Navier–Stokes equations are discretized using a sixth-order compact finite difference method. The eigenvalue problem of Eq. (6) is solved using an in-house code based on SLEPc and SuperLU_Dist [19]. The calculated eigenspectrum associated with β = 0 is plotted in Fig. 12. The horizontal axis represents the modes frequency, and the vertical axis shows their corresponding growth rate. The dashed line indicates the marginally stable condition.

Fig. 12
Fig. 12
Close modal

There are a few modes with positive growth rates. The stationary modes, with Re{ω} = 0 are indicative of unstable base flow, which is common in the cases with global instability that use RANS base flows [18]. These modes cannot help with designing active flow control techniques. The non-stationary unstable mode can be deemed a spurious mode appearing as an artifact of the small domain size that was incorporated to reduce the computational burden of the stability analysis.

The modes with Im{ω} < 0 that are closer to the dashed line are more unstable. Here, the most unstable mode is selected as the target mode and the associated pressure eigenfunctions are plotted in Fig. 13(a). This figure shows the formation and spatial growth of wave packets very close to the wall in the aft portion of the hump which can be interpreted as the dominant separating mode. The adjoint modes (left eigenvectors) are calculated and shown in Fig. 13(b) to find the most sensitive area to control this mode. Then, the sensitivity map is found by
$q*=‖q^q^†‖‖q^‖‖q^†‖$
(7)
which is plotted in Fig. 13(c). There are two areas wherein the target mode has the highest sensitivity: one is near separation, and the other is closer to the reattachment point, and both can potentially be targeted for flow control. However, the goal of this section is to find the location that can serve as the optimal point for placing the active flow control at a wide range of Mach numbers and Reynolds numbers. The separation point has a weaker dependency on these two quantities. Moreover, as discussed earlier, RANS, employed to calculate the base flow for the present sensitivity analysis, fails to identify the reattachment point while accurately predicting the separation point. Therefore, the target location to control the separation mode is placed near the separation point and is marked with the arrow in Fig. 13(c).
Fig. 13
Fig. 13
Close modal

The angle of the injection port is chosen to be 80 deg off normal to the surface (10 deg from tangent or 5.8 deg below horizontal) as suggested by Wang et al. [21].

Festo high-speed actuation valves are used to impart the driving pulsation of the flow control system due to the high flowrate and wide range of actuation frequencies (0–1000 Hz). The geometry of the flow control port is sized to adequately pass the Festo valve’s nominal flow rates and have a smaller cross-sectional area than all supply tubing such that any choking occurs in the injection port. The result is a rounded slot with a height of 1 mm and an overall width of 2.7 mm. Fifteen ports are placed 6.35 mm apart centered on the mid-plane of the wall-mounted hump.

A pulsation circuit using MOSFETs as switches to control the trigger signal is used to actuate the Festo valves. The trigger signal is a square wave provided by an Arduino Mega 2560 at the desired frequency. The circuit is shown in Fig. 14.

Fig. 14
Fig. 14
Close modal

## Controlled Flow

Controlled flow was investigated in two phases, first with steady flow injection and then with pulsated flow injection, both experimentally and numerically. The experimental Cp distribution of the flow surface can be extracted for both; however total pressure losses cannot be recovered from the experiment as there is no downstream total pressure measurement.

For each preliminary Reynolds number analyzed, the 2D simulation of slot control shows that the injected flow forms a jet that adheres to the surface of the wall-mounted hump. As the injected pressure increases compared to the inlet of the test domain, the jet becomes stronger and a lambda shock forms and moves progressively downstream and grows in strength as shown in Fig. 15. One observes the abatement of the separation bubble and emergence of a stable flow structure as the total pressure ratio of flow injection to inlet (PR) increases. A separation region just upstream of the injected jet also becomes more prevalent as the strength of the injected flow increases.

Fig. 15
Fig. 15
Close modal

Figure 16 depicts the mass-weighted pressure losses as a function of injection pressure ratio. A local minimum occurs near a total pressure ratio of 1.18. The four Reynolds numbers compared with an inlet Mach number of 0.5 all condense to the same trend, due to the small sensitivity of separated extent on Reynolds number. This chart shows that an optimal pressure ratio exists for slot injection. This optimal pressure differs from those found via experimentation as the test article uses discrete ports for flow injection.

Fig. 16
Fig. 16
Close modal

For injection pressure ratios beyond 1.2, Fig. 16 shows increasing pressure loss due to the growing size and strength of the shock. Additionally, the larger pressure in the port no longer contributes to significantly reducing the size of the SB.

### Discrete and Slot Injection.

The 3D domain is modeled using unsteady Reynolds-averaged Navier–Stokes (URANS) to assess steady injection compared to 2D numerical findings. A greater PR is needed to eliminate the separation bubble in 3D due to the change from slot injection to the discrete port injection of the experiments. The discrete port injection adds less momentum to the flow for the same PR due to the decrease in effective injection area and internal flow topologies as will be later discussed. This comparison of 2D and 3D is represented in Fig. 17, where the same PR is only effective in mitigating the separated region in the 2D RANS.

Fig. 17
Fig. 17
Close modal

In addition, the 3D numerical results show that the flow from the port is less likely to adhere to the surface of the hump than the 2D simulations suggest, implying the effectiveness of injection depends highly on the internal flow structures and losses within the injection port. Figure 18(c) shows the contours of Mach number alongside the midplane. A small recirculation is observed as a result of the change of direction of the flow in the injection port due to the elbow. In these regions, a large momentum loss deviates the flow, reducing the effective area. Near the interface, a small separation in the interior zone also occurs.

Fig. 18
Fig. 18
Close modal

However, a shock similar to that observed in the 2D RANS (note the lambda shock in Fig. 18(b)) and a small separation at the upstream corner of the injection port (better seen in Fig. 18(a)) coupled with the injection emerge. This time there is an additional shock structure within the flow emerging from the injection ports, whose throat is shown in Fig. 18(c). The lambda wave is only instantaneously formed upstream of the port and it does not appear in time-averaged variables.

After the elbow in the flow injection port, a secondary vortical structure can be observed on the duct. Looking at Fig. 18(d), two counterrotating vortices are shown that propagate streamwise through the main domain. These vortices imply a considerable reduction in the total pressure, only the relatively small region near the sides where the loss is negligible.

The effectiveness of the injection can be seen in Fig. 18(a) with the average total pressure where the separated flow is mixed with the high pressure injected one, reenergizing the region and reducing considerably the separation.

A similar topology can be observed through the schlieren imaging of steady injection. 10.441 s of images are taken at a frame rate of 2000 frames/s focused on the port (shown in Fig. 19). The first second of a given experiment is marked by a startup transience linked with supersonic flow of the 3D hump that quickly returns to transonic. Throughout this start-up transience, the port is overpowered by the SB and eventual formation of the injection leading edge shock, as shown in Figs. 19(b) and 19(c). Then a second shock forms with a separation that starts downstream and migrates towards the crest (Fig. 19(d)). This separation is affected in some way by the injection. Once the supersonic separation is upstream of injection ports (Fig. 19(e)) the inlet Mach begins to decrease, and the injection leading edge shock disappears. At around 0.85 s, the strong shocks near the crest no longer form and the transonic SB begins to interact with the imposed steady injection (Fig. 19(f)) which weakens its strength relative to the shear layers that had previously formed. After approximately 4 s, the streams from the injection ports have grown in relative strength, indicated by the visible plume length, but are coupled with a lambda shock and upstream separation (Fig. 19(i)). Figure 19(j) shows a typical time snap from between 5 s and 10 s. The flow from the injection port remains stable with a small separation appearing sporadically upstream but being eliminated by the injected mass flow. These images make the effectiveness of steady injection clear. The injected flow can interact with the separation through transient processes to enforce favorable flow topologies.

Fig. 19
Fig. 19
Close modal

The pressure coefficient comparison between the CFD cases and the experiments for Re = 3.5 × 106 m−1 and Mach 0.575 is conducted in Fig. 20. Experimentally, the length of the separation bubble is decreased from the base flow to the lowest PR case. Then, as PR is increased the main bubble is eliminated, but a smaller one appears upstream the previous one. However, this new separation is overlaid by the injection flow with no further consequences (discussed in Fig. 18(c)). Furthermore, as already discussed in Fig. 16 for the 2D RANS case, having large pressure ratios is not beneficial as no relevant improvement is obtained.

Fig. 20
Fig. 20
Close modal

The RANS result, as already discussed, overpredicts the length of the separation, as such, the same PR is not as effective as in the experiments, although still improves the base case.

The required injection PR is a function of the length of the separated region. When looking at a constant Reynolds numbers at varying Mach numbers, due to the greater sensitivity of SB length to Mach, it becomes clear that the larger the separated region the greater the PR required to mitigate separation.

The length of the separation bubble for different Mach numbers and the PR that removes the separation is summarized in Table 5. Qualitatively this effect is observed in Figs. 19(g)19(j). The Mach number drops overtime during a blow-down experiment. Similarly, the extent of the separation shrinks, and the injection overpowers the separating flow and forces sustained attachment. Before sustained reattachment is achieved, there is some unsteady interaction between the injected flow and the separating layer, but the separation is not mitigated.

Table 5

Separation bubble length for base case and optimal PR for different experimental cases

Name baseRe (m−1)M (–)SB (m)PR opt. (–)
H-04-3813.9 × 1060.6000.1362.54
H-04-5483.9 × 1060.5750.1171.71
H-04-7083.8 × 1060.5500.1151.64
H-04-8723.8 × 1060.5300.1061.61
Name baseRe (m−1)M (–)SB (m)PR opt. (–)
H-04-3813.9 × 1060.6000.1362.54
H-04-5483.9 × 1060.5750.1171.71
H-04-7083.8 × 1060.5500.1151.64
H-04-8723.8 × 1060.5300.1061.61

### Pulsated Injection.

Computationally, flow injection was pulsated in the 2D domain to assess slot injection. The Festo valves used to actuate the flow experimentally impart a 50% duty-cycle square wave. To approximate this waveform, a sine wave with an offset and amplitude of half the mass flow used for a given pressure ratio is used to model the effects of unsteady injection.

Using the 2D RANS case, frequencies of 50, 200, 500, and 750 Hz were investigated numerically to determine an effective forcing action to mitigate the separation. The time-averaged pressure along the bottom wall has little additional change beyond frequencies of 200 Hz, shown in Fig. 21. This corresponds to half the shedding frequency identified in Sec. 3.3. The plateau observed in the Cp contour of the 50 Hz pulsation indicates that the separation bubble is not mitigated in the time-averaged sense.

Fig. 21
Fig. 21
Close modal

It is observed that in the 50 Hz injection case the flow field alternates between the fully controlled and fully separated states. Frequencies beyond 200 Hz do not exhibit this alternation and instead force the shedding of the separation along the surface of the hump. Figure 22 confirms the flow topology for two half-period events, at 50 Hz and 500 Hz pulsated injection.

Fig. 22
Fig. 22
Close modal

Experimentally, a range of frequencies between 50 Hz and 400 Hz pulsated injection was applied at different total pressure ratios to assess the SB response. Figure 23 shows the experimental Cp profile along the bottom wall to finally demonstrate the efficacy of pulsated injection. All frequencies explored contributed to a decrease in the size of the separated region in a time-averaged sense. Frequencies above 150 Hz begin to have extremely diminishing returns and frequencies above 200 Hz appear to have little additional benefit, but do not show adverse effects beyond the energy required to pulsate the injection. Pulsation does not increase the effect of an injected PR, but instead reduces the mass flow requirement for a given injection pressure.

Fig. 23
Fig. 23
Close modal

## Conclusions

The efficacy of steady and unsteady flow injections in the abatement of flow separation is investigated in a blow-down test facility and with numerical simulations. The time-average response of a separation bubble is assessed with wall pressure taps, total pressure, and temperature probes, while unsteady features are investigated with schlieren imaging in the blow-down facility. The open geometry wall-mounted hump exhibits both Reynolds lapse behavior and is susceptible to separation growth with increasing inlet Mach numbers.

Numerical studies of transonic separation overpredict the extent of the separated region of the flow field. However, high fidelity simulations exhibit similar flow topologies observed during experimental blowdowns, such as wavelets and instantaneous lambda shocks.

Steady injection is shown to be an effective strategy in separation abatement, both numerically and experimentally. Though numerical studies overpredict the extent of the separation experimentally observed, similar reductions in separation size can be seen for identical injection quantities. Consequently, global stability analysis is shown to be an effective method to locate flow control features. 2D simulations underpredict the required pressure needed to eliminate the separation bubble. The ports are modeled as slots, instead of discrete entities, and the three-dimensional flow topology within the flow control port cannot be modeled in 2D. It can also be seen that the required pressure to mitigate separation grows with the extent of the separated region, due to the increasing momentum deficit associated with larger separated regions.

Likewise, unsteady flow injection is shown to be effective in abating separation bubbles. Though 2D numerical studies do not provide a range of appropriate pressures for injection, they aid in identifying effective frequencies of control. There exists an upper frequency limit, where the pulsation of the injected flow at frequencies higher than the limit no longer provides additional benefit. It is unclear whether there is an associated penalty of pulsating faster than said limit other than the energy required to achieve the pulsation.

## Acknowledgment

This research is financed by the Air Force Office of Scientific Research, Air Force Material Command, USAF under Award No. FA9550-19-S-0003 Amendment 001.

Federico Lluesma-Rodríguez was partially funded by Ministerio de Ciencia, Innovación y Universidades under the grant “Doctorandos Industriales” number DI-17–09616.

The authors also thank the US Department of Energy for the appointment of Professor Paniagua to the Faculty Research Participation Program at the NETL.

Numerical Simulations were performed on Stampede2 cluster at TACC through the Extreme Science and Engineering Discovery Environment (XSEDE) via the allocations CTS200027.

The authors are indebted to Luis Zarate-Sanchez for his tireless efforts in configuring the control system for the Festo valves, without whom the concluding experiments were not possible. The authors would also like to thank Nathaniel Kiefer, Dr. James Braun, Lakshya Bhatnagar, and Michael Butzen for their help in the test program, and measurement techniques.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

e =

relative error

f =

frequency of pulsated injection

r =

grid size ratio

s =

curve length

t =

time

K =

re-laminarization limit

L =

length of domain/test article

P =

pressure

T =

temperature

U =

freestream velocity

X =

X position

Y =

Y position

Cf =

friction coefficient

Cp =

pressure coefficient (Pt,inPs,local)/(Pt,inPs,exit)

GCI =

grid convergence index

PR =

total pressure ratio (Pt,c/Pt,in)

SB =

separation bubble

λ =

τw =

wall shear stress

ν =

dynamic viscosity

## Subscripts

a =

approximate

c =

injected flow control

s =

static quantity

t =

total quantity

in =

inlet quantity

exit =

quantity at the farthest downstream measurement

ext =

extrapolated

local =

quantity at specific location

max =

maximum

relam =

relaminarization

sep =

separated

sex =

static exit

tin =

total inlet

tot =

total (in terms of length)

θ =

momentum boundary layer thickness

### Appendix A: Uncertainty Quantification

Cp, inlet Mach, inlet Reynolds /m, and PR are discussed, all of which were derived from measurements taken during the experimental campaign. To inspire confidence in the derived quantities, their uncertainties were quantified using the method laid out by Moffatt [22]. The lowest pressure and temperature blow-down (N-15-772) was chosen to tabulate these uncertainties as the relative errors in total temperature and pressure would be highest. The crest is chosen as the location to tabulate errors in Cp as the magnitude of the pressure at the crest is the lowest in magnitude most often, due to the elevated velocity. Table 6 shows the uncertainty in Cp, inlet Mach, inlet Reynolds /m, and PR respectively. The tabulated values are a conservative estimate for the errors in all derived quantities shown.

Table 6

Tabulated uncertainties in derived quantities based on root mean square of error contributed from each measured quantity

QuantityMean valueUnitsRelative uncertainty in % mean95% band +/−
P0,inlet32,122.9Pa0.62%
Ps,local13,852.8Pa0.36%
Ps,exit26,370.5Pa0.19%
Ps,inlet25,673.5Pa0.19%
γ1.43.57%
T0,inlet283K1.06%
M0.5751.26%0.01
Re/m3,823,881m−11.16%87021
Cp,crest3.181.43%0.09
PR2.700.47%0.02
QuantityMean valueUnitsRelative uncertainty in % mean95% band +/−
P0,inlet32,122.9Pa0.62%
Ps,local13,852.8Pa0.36%
Ps,exit26,370.5Pa0.19%
Ps,inlet25,673.5Pa0.19%
γ1.43.57%
T0,inlet283K1.06%
M0.5751.26%0.01
Re/m3,823,881m−11.16%87021
Cp,crest3.181.43%0.09
PR2.700.47%0.02

All relative uncertainties are below 2%, so a more rigorous approach to quantify uncertainties was not attempted. Charts throughout this paper did not show error bars to increase legibility.

### Appendix B. Boundary Layer Status

As an additional reference to the status of the boundary layer prior to separation, boundary layer profiles are plotted in Fig. 24.

Fig. 24
Fig. 24
Close modal

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