## 4.1. Laboratory Setup and Analysis

Experiments investigating the impact of boundary actuation at various patterns and frequencies on near boundary velocity and turbulence were set up at the flowSiTTE flow channel, the laminar-to-turbulent flow tank of the Simulated Turbulence and Turbidity (SiTTE) facility at NRL SSC. The SiTTE facility consists of a large Rayleigh-Bénard convective tank, and the smaller flow channel with a working section of 0.6 m by 0.2 m by 0.2 m, capable of flow speeds up to 1.35 m/s. A detailed description of the SiTTE convective laboratory and companion numerical tank can be found in [10]. A deformable membrane system, a 5 x 5 array of voice coil motors (VCM) overlaid by a flexible polydimethylsiloxane (PDMS) membrane was integrated in the tank lid of the working section of the laboratory flow channel (Fig. 1, top).

The mechanical structure of this 5 x 5 VCM-actuated membrane assembly is shown in Fig. 1 (bottom). Twenty-five VCMs (NCM01-04-001-2IB, H2W Technologies, Inc.), which can be individually controlled, are clamped in a 3D-printed frame (using Stratasys J750 polyjet 3D-printer with VeroClear materials) with the motors’ shaft heads embedded in the PDMS membrane (about 4.5 mm thickness; Young’s modulus of about 240 kPa), forming a firm connection even under high-frequency vibration. Details of actuator control and assembly can be found in [14] and [15], respectively.

The array of 25 actuators was implemented in the lid for the flow channel working section (Supplementary material 2). The membrane covering the actuator array is in direct contact with the flow and can be controlled to deform in various wave-like patterns and at various control frequencies up to 10 Hz. An external control board connected to an embedded computer (Raspberry Pi 4 Model B, 4GB) via I2C communication, runs the controller script with a Python GUI to facilitate the switching between manual or automatic operation modes.

Experiments were performed in the flow tank for a range of flow speeds and membrane actuation patterns, and velocity data were collected with Particle Image Velocimetry (PIV). Waves were generated to align with the flow direction (streamwise) and in the spanwise, i.e. transverse or across-flow direction. Actuator control oscillating frequencies were set at 2 Hz, 4 Hz, and 10 Hz, corresponding to wave frequencies of 1 Hz, 2 Hz, and 5 Hz, respectively. However, the frequency setting translated to somewhat slower pulsation frequencies for the waves that were ultimately generated on the membrane. Thus, with an actuation setting for a wave with a wave frequency of f = 5 Hz, the frequency signal observed at measurement points in the flow near the actuated membrane was a frequency of f = 3 Hz. This value is used for the analysis presented here to calculate the phase-average for the velocity decomposition as described below. The control in this data set was taken with the actuator array in its resting state.

Velocity data were collected with PIV for a range of flow speed conditions, here we report measurements at “fast”, U0 = 0.55 m/s and “slow”, U0 = 0.09 m/s, free-stream velocities. These correspond to the slowest steady flow speed we can achieve in the tank, and an intermediate speed at which turbulence is more developed. The corresponding flow Reynolds numbers \(Re= \frac{UL}{\nu }\) are *Re**slow* = 20,000 and *Re**fast* = 110,000, respectively, where U is the flow speed, L is the length scale taken as the tank height, and *ν* is the kinematic viscosity of water. In PIV, the flow is seeded with small buoyant particles that follow the flow. Light sheet optics on a dual-head laser fan out the laser light into a sheet that illuminates the particles in the measurement plane. A CCD (charge-coupled device) camera takes image pairs of the illuminated particles at high frequency. Velocities can then be inferred by correlating the movement of particles between successive images. Figure 11 shows the working section of the flowSiTTE tank with the PIV setup: the laser light sheet is directed upwards, in a plane parallel to the flow direction, and intercepting the top of the tank lid, where the actuated boundary is installed. The CCD camera sits perpendicular to this setup, allowing it to take images of the measurement plane. The laser is a Litron ND:Yag laser, dual cavity, 145 mJ per cavity, with a 15 Hz maximum frequency per cavity, with a wavelength of 532 nm. A light guide arm directs the light sheet into the tank, parallel to flow direction, and intersecting the tank lid and deformable boundary (Fig. 11, right). The water in the tank was seeded with silver-coated glass particles (10 µm diameter). The sampling frequency was set to 15 Hz, with an interval of 5.5 ms (slow flow) and 0.8 ms (fast flow) between laser pulses to record double frame image pairs. 1000 image pairs were collected per experiment with an image size of 2352 x 1768 pixels, with a FlowSense EO 4M-41 camera. The spatial resolution of the derived velocity field varies, as the post-processing algorithm (Dynamic Studio 6.1.3) adapts the PIV interrogation area size depending on seeding density and velocity gradients, but is on the order of mm. Here, the resolution of the resulting gridded PIV velocity field is dx = dy = 0.58 mm for a field of 293 by 224 points. Figure 12 shows the Field of View (FOV) of the PIV camera at room light, and illustrates that the measurement area captures part of the deformable boundary, as well as the trailing edge behind the actuated membrane. The red and black lines in Fig. 12 indicate the locations of profile data shown in the results section of this paper. The first profile location is directly under the actuated membrane (x = − 47 mm, in PIV coordinates) and the second location is in the wake of the actuated membrane (x = 25 mm), respectively. While turbulence can be observed in the boundary layer for either flow condition, neither appears to be fully developed turbulence. This is indicated by the absence of a clear log layer in the boundary layer velocity profile, in particular for the slower flow condition (see Fig. 2). Velocity profiles as shown in Figs. 2 and 3 were averaged over all realizations (N=1000) and normalized with the respective free-stream velocity U0, whereas the distance from the wall, y, was normalized by the channel height H = 0.2 m.

To quantify the impact of actuation on the flow field, we calculate the phase-averaged velocities, Reynolds stresses, and the non-dimensionalized viscous shear stress, following [9]. The velocity U was decomposed into a mean velocity \(\stackrel{-}{U}\), a wave component \(\tilde{u}\), and the fluctuating component \({u}^{\text{'}}\):

$$U= \stackrel{-}{U}+\tilde{u}+{u}^{\text{'}}$$

1

Fluctuating velocity components, \({u}^{\text{'}}\)and \({v}^{\text{'}}\), for Reynolds stresses, are calculated by subtracting the phase-averaged fields from the instantaneous velocity. For this calculation, and in the laboratory experiments, *u* is the horizontal, along-tank velocity, and *v* the vertical velocity. The prime denotes the fluctuating component. The Reynolds stress term is calculated as \(\stackrel{-}{u\text{'}v\text{'}}\), where the overbar denotes the time average. To calculate the profile of the viscous shear stress \({\tau }_{v}\), the non-dimensionalized viscous shear stress is defined as follows:

$$\frac{\nu }{{u}_{*}^{2}}\frac{dU}{dy}$$

2

Here, ν is the kinematic viscosity of water and \({u}_{*}\)is the friction velocity, defined as \({u}_{*}=\sqrt{\frac{{\tau }_{w}}{\rho }}\), and estimated to be \({u}_{*}\)= 0.02 m/s, an estimate confirmed by the numerical model. τw is the stress near the wall, and ρ is the water density. In Fig. 6, it is plotted against the normalized wall distance *y**+*, defined as \(\frac{yu*}{\nu }\).

To complement the laboratory experiments and support interpretation of results, a numerical model was set up to emulate the laboratory tank, as was done previously for the convective SiTTE tank [10]. The setup of the numerical experiments, using a Computational Fluid Dynamics (CFD) large-eddy-simulation (LES) model, is described below.

## 4.2. Numerical Model

The working section of the flowSiTTE tank served as the fluid domain for a numerical tank, emulating the laboratory setting. The 3D model was implemented using a Large Eddy Simulation (LES) approach, with the Smagorinsky sub-grid-scale (SGS) model [16]. The numerical model solves the filtered Navier-Stokes equations along with a turbulent viscosity term, and the continuity equation for conservation of mass. Discretization is based on the finite-element method [17]. Simulations were performed at two mesh resolutions, as shown in Fig. 13. Exploration of the parameter space was performed with the domain and mesh resolution in Fig. 13, left, key cases were solved with increased boundary layer mesh resolution (Fig. 13, right).

Domain dimensions are 0.8 m x 0.2 m x 0.1 m, in x, y, and z, respectively. A 2 mm lip is located 0.2 m from the inlet to mirror the lip present when the flow enters the test section of the flowSiTTE tank. The bottom face of the mesh was set as symmetry plane. The geometry past the lip is designed to match the 0.6 m x 0.2 m x 0.2 m dimensions of the flowSiTTE test section. The top face of the geometry is a no-slip boundary, all other boundaries are free-slip, to reduce computational requirements and wall effects. The mesh is finely resolved at the tank lid, with a boundary layer prism mesh extending approximately 3 mm from the tank lid and approximately 700,000 elements in the domain. A Moving Mesh boundary condition was implemented in the top face to simulate the actuated boundary used in the lab (Fig. 14). The boundary deformation is implemented as a wave with wavelength *λ*, amplitude *a*, and wave frequency *f*. In the results shown here, all wave patterns were standing waves oriented in the streamwise direction. All numerical simulations were done at “fast” free stream velocity U0 = 0.55 m/s to allow for a transition to turbulent flow in the numerical model. For a more direct comparison to the laboratory experiment, we ran a “lab actuation” case with a wavelength *λ* = 6.7 cm, amplitude *a* = 2.5 mm, and wave frequency *f* = 3 Hz, with increased boundary layer resolution (1.2 million cells).

Figure 14 provides a detailed look at the model geometry, locations of actuated boundary and probe locations. To the bottom left of the image is the inlet, which has a normal inflow velocity of U0 = 0.55 m/s, and to the upper right is the outlet. A lip with height of 2 mm extending from the tank top wall is visible 0.2 m from the inlet, running across the tank lid in the spanwise direction. The actuated region is shown in blue, and covers a 0.1 m x 0.1 m face that starts 0.2 m from the lip (0.4 m from the inlet). Boundary actuation is confined to this area. Probe locations include a cross-sectional plane, normal to the flow direction and located at the end of the actuated region (yellow), as well as a vertical profile centered in the region of actuation, indicated in red. Another profile location (green) is located 2 cm downstream of the first profile. The “wake” region (purple) is a 0.1 x 0.1 m segment directly behind the actuated region.

To quantify the impact of boundary actuation on turbulent velocity components as shown in Fig. 8, the velocity variance of the model fields was calculated as half the sum of the velocity component variances

$$k=\frac{1}{2}( \frac{\sum {\left(u- \stackrel{-}{u}\right)}^{2}}{N}+ \frac{\sum {\left(v- \stackrel{-}{v}\right)}^{2}}{N}+ \frac{\sum {\left(w- \stackrel{-}{w}\right)}^{2}}{N} )$$

3

Here, *u*, *v*, and *w* are the velocity components in the *x*, *y*, and *z* direction, respectively, and *N* is the number of time steps over which the summation is taken. Velocity data were collected 200 times from 3–5 seconds at the steady state (i.e., at a sampling time step Δt = 0.01 s). Average velocity profiles were calculated for direct comparison to laboratory results (as in Figs. 9, 10).