Acceleration scaling for transverse momentum injection to control drag in a turbulent flow

Abstract

In one example, an active pipe enhances fluid flow by injecting momentum into a turbulent boundary layer. The active pipe includes a pipe section for fluid transport, an actuator to azimuthally oscillate the pipe section, and a controller to operate the actuator based on a non-dimensional acceleration scale multiplier. The non-dimensional acceleration scale multiplier is used to select a target non-dimensional actuation velocity and / or non-dimensional actuation period, based on an identified friction velocity and kinematic viscosity of the fluid.

Description

Acceleration Scaling for Transverse Momentum Injection to Control Drag in a Turbulent FlowCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application Number 63 / 518,876 filed on August 10, 2023, titled “Drag Reduction In Azimuthally Oscillating Turbulent Pipe Flow,” which is hereby incorporated by reference in its entirety. Aspects of this application can be understood in the context of and / or combined with aspects disclosed in PCT Pat. Application No. PCT / US2022 / 016560, filed on February 16, 2022, titled “In-Plane Transverse Momentum Injection To Disrupt Large-Scale Eddies In A Turbulent Boundary Layer.”BACKGROUND

[0002] Drag is a dissipative force created by fluid moving relative to an object. For example, a fluid moving through a pipe experiences a dissipative force that acts against the wall of the pipe. Similarly, a vehicle (e.g., a car, an airplane, a jet, a rocket, a boat, a ship, etc.) moving through air or water experiences a resistant force that acts against the movement of the vehicle. One component of this force is skin friction drag, which is created when fluid at the surface stops, slows down or changes direction. The skin friction drag creates a turbulent boundary layer. A turbulent boundary layer of the fluid interacting with the surface of an object includes small-scale eddies close to the surface, which develop into large-scale eddies extending farther from the surface.

[0003] The systems and methods described herein are applicable to situations in which a fluid is moving relative to a stationary object, a fluid is moving relative to a moving object, and an object is moving relative to a fluid. For example, an airplane may be described as moving through stationary or quasi-stationary air, even though the air is likely flowing with various currents (e.g., updrafts, downdrafts, wind, etc.). For the sake of brevity and clarity, some examples are described herein in the context of an object moving through a fluid. However, the corollary situations in which the fluid is, alternatively or additionally, moving relative to the object are also implied.

[0004] The turbulent boundary layer impedes the motion of the object relative to the fluid around it. The turbulent boundary layer may be characterized as including small-scale eddies and large-scale eddies. The turbulent boundary layer may include irregular fluid flows in the form of rotational vortices, irrotational vortices, currents, eddies, and other turbulent flows. Turbulentflows in the turbulent boundary layer can be generally characterized as exhibiting fluctuations in pressure and flow velocity.

[0005] For purposes of this disclosure, the term ‘small-scale eddy’ is used to describe near-wall turbulent flows with viscous length scales on the order of 100r| or less and time scales on the order of lOOr or less, as mathematically described in greater detail below. In contrast, the term ‘large- scale eddy’ is used herein to describe turbulent flows extending farther from the wall (and possibly not contacting the wall) that exist with time scales exceeding those of small-scale eddies. More specifically, a large-scale eddy is defined as having a time scale exceeding 300q’.

[0006] The ratio of inertial forces to viscous forces within a fluid moving relative to an object is referred to as the Reynolds number (Re), which is a dimensionless value. Throughout this disclosure, the term “fluid” is used to describe gases, liquids, and combinations thereof. The Reynolds number increases as the size of the object and / or the speed of the object increases relative to the fluid. The Reynolds number also increases as the fluid kinematic viscosity decreases. Accordingly, the Reynolds number associated with the flow of a liquid through a pipe increases as the flow speed increases. At a given velocity for a fluid flow, the Reynolds number is higher for fluids with relatively low kinematic viscosity and lower for fluids with relatively high kinematic viscosity. The same principles apply to the motion of objects through air and other gasses (e.g., the motion of airplanes, vehicles, turbine blades, projectiles, rockets, missiles, and the like), as well as the motion of liquids relative to an object (e.g., a submarine moving through water or oil moving through a pipe).

[0007] As a flow over a surface becomes turbulent, a turbulent boundary layer is formed between the surface of the object and the flow far away from the surface (where the relative velocity of the fluid is at its free-stream value (A). The large-scale eddies and the small-scale eddies in the flow contribute to the skin friction drag that slows the flow of the fluid and / or the movement of the object through the fluid. The local drag force acting on the surface (i.e., the “wall”) per unit area is the wall stress, TW. The wall stress has a time-average or mean value of and a fluctuating value

[0008] As previously described, the turbulent boundary layer includes turbulent flows of various sizes that can be classified as small-scale eddies or large-scale eddies. The smallest eddies have a characteristic length scale given by if = v / ur, where v is the fluid kinematic viscosity and uTis the friction velocity. This is often called the viscous length scale. The friction velocity, uT, canbe expressed as uT= , where p is the fluid density. The smallest eddies have a characteristictime scale q ’, which is defined as r = Accordingly, the length scale, , and the time scale, ’, UT of the smallest eddies can be characterized as:

[0011] The largest eddies in the turbulent boundary layer have a characteristic length scale, p0, equal to the boundary layer thickness, 8, and a characteristic time scale, ifl, equal toIn many common instances relating to practical applications of the presently described systems and methods, the friction velocity, uT, is approximately 20-40 times smaller than the free-stream value, Um.

[0012] In a turbulent boundary layer, the friction Reynolds number, ReT, represents the ratio of the viscous length scale of the largest eddies to the viscous length scale of the smallest eddies and is thus expressible as:

[0013] Ret = 8iit / v. Equation 3

[0014] Accordingly, the range of scale between the large-scale eddies and the small-scale eddies in a turbulent boundary layer increases as the friction Reynolds number, Ret, increases. By way of example, the friction Reynolds number, ReT, associated with the flow of air over an airplane fuselage may be approximately 100,000, while the friction Reynolds number, Ret, associated with a fluid flowing through a large pipeline may exceed 1,000,000. As the friction Reynolds number, Ret, increases, the relative contribution of the large-scale eddies to the production of turbulence, and therefore drag, increases.

[0015] Attempts to reduce skin friction drag have focused on mitigating or modifying small- scale eddies in flows with friction Reynolds numbers, Ret, less than 1,000. Some of these approaches are described in the following publications, each of which is hereby incorporated by reference in its entirety to the extent allowed by law and assumed to be understood by one of skill in the art: Choi et al., Drag reduction of turbulent pipe flows by circular-wall oscillation, Phys. Fluids 10, 7-9 (1998) (hereinafter “Choi”); Marusic et al., An energy-efficient pathway to turbulent drag reduction Nat. Commun. 12, 1-8 (2021) (hereinafter “Marusic”); Coxe et al., Reynolds number effect on drag reduction in pipe flows by a transverse wall oscillation, available as anarXiv preprint at arXiv:2312.03232 (2023) (hereinafter “Coxe”); Batchelor, An Introduction to Fluid Dynamics (Cambridge Mathematical Library), Cambridge University Press (2000), doi: 10.1017 / CB09780511800955; Gatti et al., Reynolds-number dependence of turbulent skinfriction drag reduction induced by spanwise forcing, Journal of Fluid Mechanics (2016), vol. 802, pp. 553-582; Corke et al., Active and Passive Turbulent Boundary layer Drag Reduction, AIAA Journal (October 2018), Vol. 56, No. 10, pp. 3835-3847; Kline et al., The structure of turbulent boundary layers, Journal of Fluid Mechanics (1967), Vol. 30, pp. 741-773; Mathis et al., Estimating wall-shear-stress fluctuations given an outer region input, Journal of Fluid Mechanics (2013), Vol. 715, pp. 163-180; Marusic et al., Predictive model for wall-bounded turbulent flow, Science (2010), Vol. 329(5988), pp. 193-196; Panton, Overview of the self-sustaining mechanisms of wall turbulence, Prog. Aerosp. Sci. (2001), Vol. 37, pp. 341-383; Smith et al., The characteristics of low -speed streaks in the near-wall region of a turbulent boundary layer, Journal of Fluid Mechanics (1983), Vol. 129, pp. 27-54; Smits et al., High Reynolds Number Wall Turbulence, Annu. Rev. Fluid Meeh. (2011), Vol. 43, pp. 353-375; Alfredsson et al., Large-Eddy BreakUp Devices - a 40 Years Perspective from a Stockholm Horizon, Flow Turbulence Combust (2018), Vol. 100, pp. 877-888; Garcia-Mayoral et al., Drag reduction by riblets, Phil. Trans. R. Soc. A (2011), Vol. 369, pp. 1412-1427; Schoppa et al., A large-scale control strategy for drag reduction in turbulent boundary layers, Physics of Fluids (May 1998); Vol. 10(5), pp. 1049-51; and Gouder, Turbulent Friction Drag Reduction Using Electroactive Polymer Surfaces, Doctoral thesis, Imperial College, May 2011. Additional references relating to drag reduction include U.S. Patent No. 8,074,938 titled “Active control of a body by altering surface drag,” by Hyde et al.; U.S. Patent No. 8,074,939 titled “Active Control of Surface Drag,” by Hyde et al.; and U.S. Patent No. 9,002,484 titled “System and method for deforming surfaces,” by Hyde et al., collectively referred to as “the Hyde Patents.”

[0016] Some of the disclosures identified above suggest drag reduction techniques in which components on a surface, or portions of the surface, are moved up and down at high frequency to disrupt small-scale eddies. The up-and-down movement described in these disclosures is not parallel or co-planar to the surface. Past work has suggested or assumed that drag reduction scales with (i) the velocity amplitude of the motion and (ii) the period of the motion, in the context of the Reynolds number of the fluid.

[0017] Accordingly, the prior focus has been on controlling the injection of momentum in terms of the velocity amplitude and the period. The existing literature does not suggest, and in some instances even teaches away from, large-scale eddy actuation by controlling the acceleration of motion used to inject momentum parallel to the surface and transverse (e.g., spanwise) to the direction of the fluid flow (e.g., the streamwise direction). Moreover, existing literature does not contemplate high-acceleration movement using relatively high-frequency and / or relatively small displacement actuation for transverse momentum injection.BRIEF DESCRIPTION OF THE DRAWINGS

[0018] FIG. 1A illustrates an example drag reduction system to reduce drag in a pipe by azimuthally oscillating a section of the pipe, according to one embodiment.

[0019] FIG. IB illustrates a portion of a pipeline shell and a pipe section formed as a sleeve with a slit to be inserted therein, according to one embodiment.

[0020] FIG. 1C illustrates the sleeve inserted within the pipeline shell, according to one embodiment.

[0021] FIG. 2A illustrates graphs of the drag reduction attained in a pipe flow with respect to the non-dimensional velocity amplitude, according to one embodiment.

[0022] FIG. 2B illustrates graphs of the drag reduction attained in a pipe flow with respect to the non-dimensional period of oscillation, according to one embodiment.

[0023] FIG. 3A illustrates a graph of the drag reduction in a pipe flow with respect to the non- dimensional acceleration for a range of Reynolds numbers, according to one embodiment.

[0024] FIG. 3B illustrates a graph of the drag reduction in a pipe flow with respect to the non- dimensional acceleration for another range of Reynolds numbers, according to one embodiment.

[0025] FIG. 4 illustrates a graph of azimuth oscillation frequencies and oscillation distances in a pipe having a diameter of 38.1 mm for various friction Reynolds number values, / Av, of air and water, according to one embodiment.

[0026] FIG. 5 illustrates a graph of azimuth oscillation frequencies and oscillation distances in a pipe having a diameter of 381 mm for various friction Reynolds number values, Av, of air and water, according to one embodiment.DETAILED DESCRIPTION

[0027] The presently described systems and methods are applicable to a wide variety of fluids and surfaces that are in motion relative to one another. Examples of surfaces that may utilize the systems and methods described herein include, but are not limited to, fixed-wing aircraft, rotary- wing aircraft, rockets, missiles, projectiles, and the like. Additional examples include, but are again not limited to, various surfaces of a pipe, a pump, a fan, a turbine, a wind turbine, a mast, an airfoil, a hydrofoil, a sail, a boat rudder, a boat hull, a rocket nozzle, and a land vehicle.

[0028] The systems and methods described herein may also be utilized to decrease or selectively increase friction within pipes or fluid vessels that operate to transport fluids, mix fluids with other fluids or substances, transfer heat from fluids, or manage chemical reactions of fluids (e.g., with other fluids and / or reactants). Examples of fluids for which in-plane, transverse momentum injection may decrease (or selectively increase) friction include, but are not limited to, air, water, gaseous mixtures, natural gas, various liquids, oil, and the like.

[0029] As described in greater detail below, a controller may control the actuation of any number of actuators, and multiple controllers and associated sets of actuators may operate in concert to achieve a target friction profile, friction reduction, or friction increase along one or more surfaces of an object that is in motion with respect to a fluid and / or a fluid that is in motion with respect to the object. Examples of suitable actuators that can be used to inject transverse momentum into the turbulent boundary layer in-plane with respect to the surface and at low frequencies to disrupt the large-scale eddies include, but are not limited to, piezoelectric actuators, electromagnetic actuators, electromechanical actuators, and dielectric-barrier discharge (DBD) devices.

[0030] In various embodiments, the controller implements an acceleration-controlled injection of momentum transverse to fluid flow. For example, an actuator may azimuthally oscillate a section of a pipe (e.g., rotate the section of pipe back and forth). Previous approaches may have focused on injecting momentum with a focus on achieving a target rotational displacement and / or a target maximum velocity. In contrast, the presently described systems and methods are based on a discovery that a key parameter to reducing (or increasing) drag is the acceleration of the spanwise motion used for transverse moment injection. For example, the presently described systems and methods contemplate higher frequency actuation of spanwise motion with relatively smallerdisplacement distances, as compared to approaches focused on the velocity amplitude of the motion and / or the period of the motion.

[0031] In some embodiments, an actuator may operate to oscillate a portion of a surface in time, t, with an oscillation velocity, m, given by:

[0032] a>s(t) = .4 sin(2ut / T) Equation 4

[0033] In Equation 4, the oscillation velocity, m, is a function of the amplitude of the spanwise velocity, A, and the period of oscillation, T. For example, an actuator may operate to oscillate a section of a pipe in the azimuthal direction. The surface motion induces a spanwise velocity component in the fluid within the pipe very near the surface. The surface motion results in a transverse injection of momentum that is well-described by a Stokes layer. The transverse momentum injection interferes with the turbulence production to reduce or otherwise modify the drag. For instance, the transverse momentum injection may interfere with the streak-like structures found close to the surface at the interface between the fluid and the inner wall of the pipe. As noted above, a system to reduce or otherwise modify drag can be greatly simplified (e.g., by two orders of magnitude) by approaching the actuation control in the context of the acceleration of motion used to inject momentum transverse to the direction of the fluid flow.

[0034] The previously unrecognized impact of the acceleration parameter in drag reduction via transverse momentum injunction can be better understood in the context of local drag per unit area. For example, the local drag per unit area, averaged across the surface or wall, TW, has a functional dependence, g}, expressible in the context of a pipe as:

[0035] TW=gl( U, R, p,v, T,A) Equation 5

[0036] In Equation 5, U represents the bulk velocity, R is the radius of the pipe, p is the fluid density, and v is the kinematic viscosity. It is appreciated that a similar functional dependence can be expressed for surfaces having other shapes (e.g., for a flat surface, a curved surface, a pipe with an elliptical cross-section, etc.). It is understood that the examples include the case of a boundary layer where the appropriate length scale is the boundary layer thickness 5 instead of the radius R. Using the example above, the drag reduction, DR, can be described as the fractional decrease in the local drag per unit area of the wall, TW, with the dimensional analysis expressible as:

[0037] DR = g2(A+, T+, ReT) Equation 6

[0038] In Equation 6, uT0= -^ / TW0 / P and ReT= RUTQ / V is the Reynold number, ReT. In many embodiments of the systems and methods described herein, the Reynolds number ReTis at least 180 and, in many cases, in the thousands, tens of thousands, hundreds of thousands, and even millions. The subscript “0” denotes a quantity in the non-actuated state over the surface. The non- dimensional velocity, A+, is equal to A / UTQ, and the non-dimensional period, T+, is equal to Tu^o / v.

[0039] The presently described systems and methods leverage a discovery that the suppression of turbulence to reduce skin friction is associated with the rate of change of the transverse shear component of injected momentum, which is typically highest during the transitional phase of oscillation. For wall oscillation, the rate of change of dcos / dy scales with A+ / T+. Accordingly, the presently described systems and methods propose the use of a non- dimensional acceleration scale multiplier, a+, where:

[0041] Previous approaches for determining the actuation motion for a given Reynold number, Re , were scaled based on the non-dimensional velocity, A+, the non-dimensional period, T+, and / or other parameters, such as the influence range of the Stokes Lyer. The complexities of the calculations to maximize drag reduction based on the non-dimensional velocity, A+and the non-dimensional period, T+, are apparent in Equation 6 above. In stark contrast, the presently described approach to determine the actuation parameters for a target drag reduction based on the non-dimensional acceleration scale multiplier, a+, results in a significant reduction in complexity. Significant drag reduction can be attained for non- dimensional periods, T+, greater than 100.

[0042] In addition, the power required to move the fluid scales approximately as (T)2'5, and so there are significant benefits to operating at relatively low frequencies (e.g., where T+> 100 or at least where T+> 80). Such low-frequency, large-period actuation provides a realistic pathway to energy-efficient drag reduction in fluids having relatively high Reynolds numbers. Actuation design and control based on a function of the non-dimensional acceleration scale multiplier, a+, serves as a powerful design tool for achieving target drag reduction (e.g., maximized drag reduction). In terms of the oscillation frequency, f+, wheref+= 1 / T+, and amplitude, d+, a+oc (f+)2d+. Additionally, (f+)2and d+have similar effects on achieving a target (e.g., maximum) drag reduction.

[0043] In one example, an active pipe includes a pipe section to transport a fluid. An actuator operates to azimuthally oscillate the pipe section to inject momentum into a turbulent boundary layer of the fluid. The momentum is injected tangentially in-plane with an inner surface of the pipe section and transverse to a flow direction of the turbulent boundary layer of the fluid, relative to the inner surface of the pipe section.

[0044] A controller may operate or actuate one or more actuators to azimuthally oscillate the pipe section with an actuation frequency, in Hz so that the period of the motion is given by T =1 / f and the angular frequency co is defined as co = 2nf rad / s. In terms of the viscous time scale, r , the period, T, can be expressed as a non-dimensional period T+(also referred to as a time scale multiplier) defined as T+17 . For any given Reynold number, Rex, Y]' is constant. As an example, if the controller is set to operate with a non-dimensional period, T , in the order of 100, the actuation period (T) is similar to the time scale of (and therefore actuates) the small-scale eddies in the flow. Similarly, because large-scale eddies have time scales exceeding 300r|’, when the controller is set to operate with a T+of more than 300, the actuation period (T) is similar to the time scale of (and therefore actuates) the large-scale eddies in the flow. According to various embodiments, the controller may be configured to operate with a non-dimensional period, T+, selected to actuate large-scale eddies and / or small-scale eddies.

[0045] It is understood that in other contexts, such as in a boundary layer, the appropriate length scale is the boundary layer thickness 6 instead of the pipe radius, and all parameters that use R would instead be defined using 5.

[0046] The controller may identify the specific fluid flow characteristics of a turbulent boundary layer of the fluid. For example, the controller may be pre-programmed with specific data or use sensor data (e.g., in real-time) to measure one or more characteristics of the fluid flow. Examples of fluid flow characteristics to be obtained from memory, a third-party server, or via sensor measurements include, but are not limited to, a mean or bulk velocity, U, (or a freestream velocity Ux), a friction velocity, uT, of the fluid, a kinematic viscosity, v, of the fluid, and friction Reynolds numbers, ReT, of the fluid. In some instances, some or all these values may be estimatedbased on known fluid types, the relative velocity of the surface and the fluid, the temperature of the surface and / or fluid, and / or the like.

[0047] The controller may calculate an actuation frequency, for transverse momentum injection along the surface (e.g., for in-plane momentum injection co-planar with the surface). In the context of an azimuthally oscillated pipe, the injected momentum is accurately described as being co-planar with the surface as the momentum is injected at each location in a direction tangent to the curved surface of the pipe (e.g., the injected momentum is “tangentially co-planar to the surface”).

[0048] The controller may actuate one or more actuators using the calculated or determined actuation frequency, to selectively increase (or decrease) the drag of the fluid on the surface of the object. The non-dimensional period, T is a function of the identified friction velocity squared, u / divided by the product of (i) the calculated actuation frequency, f and (ii) the identified kinematic viscosity, v. Thus, the non-dimensional period, 7 / is expressible as:

[0050] A controller operates the actuator with a non-dimensional acceleration scale multiplier, a+, of at least 0.001 for fluid flows having friction Reynolds numbers, ReT, greater than 180. As described above, the non-dimensional acceleration scale multiplier, a+, is a function of the non-dimensional actuation velocity, A+, divided by the non-dimensional actuation period, T+(per Equation 7 above).

[0051] In some examples, the controller operates the actuators with a sinusoidal actuation displacement. In other examples, the controller operates the actuators with a non- sinusoidal actuation displacement. For example, since acceleration is the key parameter, it may be advantageous to use a non-sinusoidal waveform (or other non-sinusoidal actuation paradigm) to increase or even maximize the cycle-integrated acceleration.

[0052] In some examples, it may be difficult to rotate the pipe itself without introducing mechanical complications or leakage of the fluid (e.g., air, water, gas, or another fluid). Accordingly, the azimuthally oscillated "pipe section” may be a sleeve positioned within an outer pipeline shell. The sleeve may be azimuthally oscillated to inject the tangentially transverse momentum for drag reduction. The sleeve may, in some embodiments, be formed as a complete cylinder (e.g., a circular or elliptical cylinder). In other embodiments, thesleeve may have a lengthwise slit to facilitate positioning the sleeve within the pipeline shell. In such embodiments, a cross-section of the sleeve may form a partial ellipse or partial circle.

[0053] Many of the examples provided herein are described in the context of drag reduction within a pipe. However, it is appreciated that the same principles can be applied to other surfaces moving within a fluid, other fluids moving with respect to surfaces, and / or combinations thereof. Without limitation, the presently described systems and methods are applicable to a wide variety of fluids and surfaces, including surfaces on an object such as a fixed-wing aircraft, a rotary-wing aircraft, a rocket, a missile, a projectile, a pump, a fan, a turbine, a wind turbine, a mast, an airfoil, a hydrofoil, a sail, a boat rudder, a boat hull, a rocket nozzle, a vehicle, a duct, etc.

[0054] As such, a more generalized turbulent boundary layer control system may include a set of actuators on the surface of an object (e.g., integrated on, within, or even forming the surface of the object). Each actuator is selectively controllable (either individually, in groups, or collectively) to inject momentum into a turbulent boundary layer of a fluid in-plane with the surface and transverse to a flow direction of the turbulent boundary layer of the fluid relative to the surface. As described herein, a controller may operate the set of actuators according to a selected non-dimensional acceleration scale multiplier, a+, of at least 0.001.

[0055] Again, the actuators may be operated with a sinusoidal or non-sinusoidal actuation displacement. In some examples, at least some of the actuators are spaced on the surface in a direction transverse to the flow direction of the fluid with a spacing distance between one- tenth and one-half of a boundary layer thickness, 6, of the flow, where the boundary layer thickness, 8, of the flow.

[0056] As described herein, a method to modify drag between a fluid and a surface may include, without limitation, identifying fluid flow characteristics of a turbulent boundary layer of a fluid flowing relative to a surface of an object having friction Reynolds numbers, ReT, greater than 180. Examples of such fluid flow characteristics include, without limitation, a mean or bulk velocity, U, (or a freestream velocity Uxa friction velocity, itT, of the fluid, a kinematic viscosity, v, of the fluid, and / or a friction Reynolds numbers, ReT, of the fluid. The method may include selecting a non-dimensional acceleration scale multiplier, a+, of at least0.001 based on the identified fluid flow characteristics of the fluid flow (e.g., estimated, measured, or calculated characteristics of the fluid flow).

[0057] As described herein, the non-dimensional acceleration scale multiplier, a is expressible as a+= vfA / u . Based on the selected non-dimensional acceleration scale multiplier, ci , the method may further include determining an actuation frequency, f and actuation velocity, A, for injecting momentum along the surface of the object and perpendicular to a direction of fluid flow relative to the surface of the object. The controller may use the determined actuation frequency, and actuation velocity, A, to actuate one or more actuators on the surface of the object. In various examples, the actuation frequency is at least 0.1 Hertz.

[0058] Many existing computing devices and infrastructures may be used in combination with the presently described systems and methods. Some of the infrastructure that can be used with embodiments disclosed herein is already available, such as processors, microprocessors, microcontrollers, computer programming tools and techniques, digital storage media, and communication links. Many of the systems, subsystems, modules, components, and the like that are described herein may be implemented as hardware, firmware, and / or software. Various systems, subsystems, modules, and components are described in terms of the function(s) they perform because such a wide variety of possible implementations exist. For example, it is appreciated that many existing programming languages, hardware devices, frequency bands, circuits, software platforms, networking infrastructures, and / or data stores may be utilized alone or in combination to implement a specific function.

[0059] It is also appreciated that two or more of the systems, subsystems, components, modules, etc., that are described herein may be combined as a single system, subsystem, module, or component. Moreover, many of the systems, subsystems, components, and modules may be duplicated or further divided into discrete systems, subsystems, components, or modules to perform subtasks of those described herein. Any of the embodiments described herein may be combined with any combination of other embodiments described herein. Many of the embodiments of the systems and methods described herein that appear to be mutually exclusive may be used in combination as weighted functions of one another and / or in primary-backup configurations in which one embodiment is used primarily, and the other embodiment is available as a backup.

[0060] As used herein, a computing device, system, subsystem, module, or controller may include a processor, such as a microprocessor, a microcontroller, logic circuitry, or the like. A processor may include one or more special-purpose processing devices, such as an applicationspecific integrated circuit (ASIC), a programmable array logic (PAL) device, a programmable logic array (PLA), a programmable logic device (PLD), field-programmable gate array (FPGA), or another customizable and / or programmable device. The computing device or controller may also include a machine-readable storage device, such as non-volatile memory, static RAM, dynamic RAM, ROM, CD-ROM, disk, tape, magnetic, optical, flash memory, or another machine- readable storage medium. Various aspects of certain embodiments may be implemented using hardware, software, firmware, or a combination thereof.

[0061] The components of some of the disclosed embodiments are described and illustrated in the figures herein. Many portions thereof could be arranged and designed in a wide variety of different configurations. Furthermore, the features, structures, and operations associated with one embodiment may be applied to or combined with the features, structures, or operations described in conjunction with another embodiment. In many instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of this disclosure. The right to add any described embodiment or feature to any one of the figures and / or as a new figure is explicitly reserved.

[0062] The embodiments of the systems and methods provided within this disclosure are not intended to limit the scope of the disclosure but are merely representative of possible embodiments. In addition, the steps of a method do not necessarily need to be executed in any specific order or even sequentially, nor do the steps need to be executed only once.

[0063] FIG. 1A illustrates an example drag reduction system 100 that azimuthally oscillates a pipe section 120 to reduce the drag of fluid flow therein, according to one embodiment. In the illustrated example, the pipe section 120 azimuthally oscillates to inject momentum into a turbulent boundary layer of the fluid in-plane with an inner surface of the pipe section 120 and transverse to a flow direction 160 of the fluid through the pipe section 120. The oscillating pipe section 120 is part of a larger pipe or pipeline 110 that is stationary. In some implementations, the entire pipeline 110 may be azimuthally oscillated and / or include multiple discrete pipe sections that can be azimuthally oscillated independently from one another or together.

[0064] In the illustrated example, the pipe section 120 is driven by a crank-slider mechanism 132 and 133. The crank-slider mechanisms 132 and 133 are connected to a timing belt 134 or chain. A motor 131 is connected to the crank-slider mechanism 132 and 133 via a T-slot joint or other interface. A controller 140 drives the motor 131, and the T-slot joint allows for adjustable amplitudes. Electronically controlled actuation components can be used to dynamically or adjustably modify the amplitude of the azimuth oscillations. The specific mechanical linkages illustrated in FIG. 1A are merely examples of possible actuator devices and associated actuator components (e.g., forming an “actuator” 130). Any of a wide variety of alternative actuators, including those described herein, may be used as part of a drag reduction system.

[0065] In one specific embodiment, measurements were made to demonstrate the feasibility and functionality of the proposed acceleration-based drag reduction using a 1 ,2-meter-long recirculating water pipe section with an inner diameter of 38.1 millimeters. In the illustrated example, the crank-slider mechanisms 132 and 133 translate linear motion to azimuthal oscillation of the pipe section 120 via the timing belt 134. In one embodiment, the length ratio of the crank to the rod connecting the crank to the slider is less than 0.05 so that the oscillation closely approximates a sinusoidal motion. In some embodiments, the controller may actuate the actuator 130 or other associated components to oscillate with a non-sinusoidal actuation displacement. In such an embodiment, the non-sinusoidal actuation displacement may maximize the cycle- integrated acceleration of the azimuthal rotation of the pipe section 120.

[0066] Measurement data is provided below for oscillation azimuthal amplitudes (d) up to 12.8 millimeters and oscillation frequencies (f= L T) up to 20 Hertz within a fully developed flow. As detailed herein, measurement data for the fully developed flow includes bulk velocities between 1.1 meters / second and 4.2 meters / second, temperature adjustments up to 57 degrees Celsius, and Reynolds numbers ranging from approximately 800 to 7000. Additional data measurements, calculations, and simulation data demonstrate the achievable drag reduction (or drag creation) of other types of fluid flows (e.g., air, gases, oil, other liquids, etc.). Moreover, the systems and methods described herein are applicable to and can be adapted for use with other surfaces and objects, such as surfaces on a fixed-wing aircraft, a rotary-wing aircraft, a rocket, a missile, a projectile, a pump, a fan, a turbine, a wind turbine, a mast, an airfoil, a hydrofoil, a sail, a boat rudder, a boat hull, a rocket nozzle, a vehicle, a duct, or the like.

[0067] According to various embodiments, the controller 140 operates the actuator 130 (e.g., via the motor 131) with a non-dimensional acceleration scale multiplier, a of at least 0.001. As previously described, the non-dimensional acceleration scale multiplier, a+, is a function of the non-dimensional actuation velocity, A+, divided by the non-dimensional actuation period, T+. As described herein, the non-dimensional actuation velocity, A+, is equal to the amplitude of the actuation velocity, A, divided by a friction velocity, uT, of the fluid, such that A+= — , and thenon-dimensional period, T+, is equal to the actuation period, T, (e.g., 1 / f) multiplied by the squareof a friction velocity, 11 , divided by a kinematic viscosity, v, of the fluid, such that T+=

[0068] FIG. IB illustrates a portion of a pipeline shell 111 and a pipe section formed as a sleeve 121 with a slit 122 to be inserted therein, according to one embodiment. The sleeve 121 can be azimuthally rotated within the pipeline shell 111 to reduce the drag of the fluid flowing therein. In some embodiments, the slit 122 may facilitate the insert of the sleeve 121 within the pipeline shell 111. The slit 122 may be larger or smaller. In some embodiments, the sleeve 121 may be externally rotated, electrically rotated, or magnetically rotated. FIG. 1C illustrates the sleeve 121 inserted within the pipeline shell 111, according to one embodiment.

[0069] FIG. 2A illustrates a graph 250 of the drag reduction on the vertical axis relative to the non-dimensional velocity amplitude, A+, on the horizontal axis for various Reynolds numbers, ReT, according to one embodiment. The graph 250 shows that the presently described systems and methods can be used to achieve a drag reduction of more than 35% for Reynolds numbers, Re. of 808 for velocity amplitudes, A+, of approximately 40. Additionally, a cross is used for data from Coxe and diamonds are used for data from Choi, for Reynolds numbers, Re , less than 1000.

[0070] FIG. 2B illustrates a graph 260 of the drag reduction on the vertical axis relative to the non-dimensional actuation period, T+, for the same set of example Reynolds numbers, ReT. Again, a cross is used for data from Coxe and diamonds are used for data from Choi, for Reynolds numbers, ReT, less than 1000.

[0071] Additionally, the graphs 250 and the graph 260 from FIGS. 2A and 2B show that the drag reduction increases monotonically with the non-dimensional velocity amplitude, A ' , and decreases monotonically with increases to the non-dimensional actuation period, T+. Neither the non-dimensional velocity amplitude, A+, nor the non-dimensional actuation period, T+, collapse the data.

[0072] The data in the graphs 250 and 260 of FIGS. 2A and 2B demonstrate that the drag reduction does not scale with the non-dimensional velocity amplitude, A+, or the non-dimensional actuation period, T+. This is especially true for the dependence on the non-dimensional actuation period, T+. As such, previous actuation control approaches that scale based on the non- dimensional velocity, A+, the non-dimensional period, T+, and / or other parameters, such as the influence range of the Stokes layer, are more complex than necessary and may not allow for the identification of the ideal actuation parameters (e.g., frequency, actuation distance, varying acceleration via non-sinusoidal actuation, etc.) to attain a target drag modification (e.g., maxim drag reduction).

[0073] As previously described, the suppression of turbulence to reduce skin friction is associated with the rate of change of the transverse shear component of injected momentum, which is typically highest during the transitional phase of oscillation. For wall oscillation, the rate of change of dois / dy in the Stokes layer scales with A+ / T+. As described herein, scaling based on the non-dimensional acceleration scale multiplier, a+, allows for a less complex and / or more accurate determination of actuation parameters to achieve a target drag reduction. For example, a maximum drag reduction for a given set of flow parameters can be achieved using Equation

[0074] FIG. 3 A illustrates a graph 350 of the drag reduction in a pipe flow with respect to the non-dimensional acceleration, a for a range of Reynolds numbers, Re , according to one embodiment. The non-dimensional acceleration, a , is expressible as or = A / T+.iand the data points in graph 350 are plotted in a logarithmic scale for non-dimensional periods, T+, greater than or equal to 100 for the range of Reynolds numbers, ReT. As illustrated, the plotted drag reduction on the vertical axis is well-correlated with the non-dimensional acceleration, a+, on the horizontal axis. A convincing collapse of the data demonstrates the superiority of scaling the actuation of transverse momentum injection based on the non-dimensional acceleration, a+.

[0075] FIG. 3B illustrates a graph 360 of the drag reduction in a pipe flow with respect to the non-dimensional acceleration, ct for another range of Reynolds numbers, Re^ according to one embodiment. As illustrated in graphs 350 and 360 of FIGS. 3A and 3B, significant drag reduction is attained for values of non-dimensional acceleration, a+, exceeding 0.01. Appreciable drag reduction is even attained for non-dimensional acceleration, a values as low as 0.001, for non-dimensional periods, T+, between approximately 80 and 130. As such, actuation design and control based on a function of the non-dimensional acceleration scale multiplier, a+, serves as a powerful tool for achieving target drag reduction (e.g., maximized drag reduction).

[0076] FIG. 4 illustrates a graph 410 of azimuth oscillation frequencies and oscillation distances in a pipe having a diameter of 38.1 mm to optimize drag reduction for various friction Reynolds number values, ReT, of air and water, according to one embodiment. For a given set of fluid flow parameters, the non-dimensional acceleration, rz+, is a function of the frequency squared and the amplitude of oscillation, a+oc ( / ?+)2* d+. As such, a large non-dimensional acceleration, a+, can be achieved with a relatively small amplitude of oscillation (e.g., small movement) at relatively high frequencies. Notably, piezoelectric oscillation devices function well at high frequencies and small movements.

[0077] The solid lines for air and water correspond to optimal values for the oscillation distance and azimuthal oscillation frequency for the non-dimensional period, T+, of 100, for various Reynolds numbers, ReT, (as labeled). The dashed lines directly above and below each respective solid line for air and water correspond to the optimal values for the oscillation distance and azimuthal oscillation frequency for the non-dimensional period, T+, values of 50 and 150 for the respective fluids. The illustrated values provide drag reduction between approximately 16-30% and correspond to a non-dimensional acceleration, a . of approximately 0.15.

[0078] As an example, the graph 410 shows that for water flowing through the pipe with a Reynolds number, ReT, of approximately 4000, an azimuthal oscillation frequency of 440 Hz should be used with a displacement distance (oscillation amplitude) of 1.1 millimeters. For air flowing through the same pipe with the same Reynolds number, ReT, of approximately 4000, an azimuthal oscillation frequency of 6200 Hz should be used with a displacement distance (oscillation amplitude) of 1.1 millimeters.

[0079] FIG. 5 illustrates a graph 510 of azimuth oscillation frequencies and oscillation distances in a large pipe having a diameter of 381 mm to optimize drag reduction for various friction Reynolds number values, ReT, of air and water, according to one embodiment. Again, a large non-dimensional acceleration, a+, can be achieved with a relatively small amplitude of oscillation (e.g., small movement) at relatively high frequencies. The solid lines for air and watercorrespond to optimized values for the oscillation distance and azimuthal oscillation frequency for the non-dimensional period, T+, of 100, for various Reynolds numbers, / / cr. The dashed lines directly above and below each respective solid line for air and water correspond to the optimal values for the oscillation distance and azimuthal oscillation frequency for the non- dimensional period, T+, values of 50 and 150 for the respective fluids. The illustrated values provide drag reduction between approximately 16-30% and correspond to a non- dimensional acceleration, a+, of approximately 0.15.

[0080] The graph 510 shows that for water flowing through the pipe with a Reynolds number, ReT, of approximately 8000, an azimuthal oscillation frequency of 18 Hz should be used with a displacement distance (oscillation amplitude) of 6.0 millimeters. For air flowing through the same pipe with the same Reynolds number, ReT, of approximately 8000, an azimuthal oscillation frequency of 250 Hz should be used with the same 6.0-millimeter displacement distance.

[0081] This disclosure has been made with reference to various exemplary embodiments, including the best mode. However, those skilled in the art will recognize that changes and modifications may be made to the exemplary embodiments without departing from the scope of the present disclosure. While the principles of this disclosure have been shown in various embodiments, many modifications of structure, arrangements, proportions, elements, materials, and components may be adapted for a specific environment and / or operating requirements without departing from the principles and scope of this disclosure. These and other changes or modifications are intended to be included within the scope of the present disclosure.

[0082] This disclosure is to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope thereof. Likewise, benefits, other advantages, and solutions to problems have been described above with regard to various embodiments. However, benefits, advantages, solutions to problems, and any element(s) that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential feature or element. This disclosure includes and encompasses at least the following claims and all possible permutations thereof.

Claims

What is claimed is:

1. An active pipe, comprising: a pipe section to transport a fluid; an actuator to azimuthally oscillate the pipe section to inject momentum into a turbulent boundary layer of the fluid tangentially in-plane with an inner surface of the pipe section and transverse to a flow direction of the fluid relative to the inner surface of the pipe section; and a controller to operate the actuator with a non-dimensional acceleration scale multiplier, ct+, of at least 0.001 for fluid flows having friction Reynolds numbers, ReT, greater than 180, wherein the non-dimensional acceleration scale multiplier, a+, is a function of a non- dimensional actuation velocity, A+, divided by a non-dimensional actuation period, T+, where the non-dimensional actuation velocity, A+, is equal to an amplitude of anactuation velocity, A, divided by a friction velocity, uT, of the fluid, such that A+=and where the non-dimensional period, T+, is equal to an actuation period, T, multiplied by the square of a friction velocity, u , divided by a kinematic viscosity, v, of the fluid, such that2. The active pipe of claim 1, wherein the non-dimensional actuation period, 7 / is at least 100.

3. The active pipe of claim 1, wherein the non-dimensional actuation period, T~, is expressible as:1 f *v— = — , where / is an actuation frequency of at least 0.1 Hertz.

4. The active pipe of any one of claims 1-3, wherein the controller operates the actuator with a sinusoidal actuation displacement.

5. The active pipe of any one of claims 1-3, wherein the controller operates the actuator with a non-sinusoidal actuation displacement.

6. The active pipe of claim 5, wherein the controller operates the actuator with a non-sinusoidal waveform for actuation displacement to maximize cycle-integrated acceleration.

7. The active pipe of claim 1, wherein the pipe section comprises a sleeve within an outer pipeline shell.

8. The active pipe of claim 7, wherein the sleeve has a lengthwise slit, such that a cross-section of the sleeve forms a partial ellipse.

9. The active pipe of claim 8, wherein the cross-section of the sleeve forms a partial circle.

10. The active pipe any one of claims 1 -3, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of: air, water, and oil.

11. A drag control system, comprising: a set of actuators on a surface of an object, wherein each actuator is selectively controllable to inject momentum into a turbulent boundary layer of a fluid in-plane with the surface and transverse to a flow direction of the fluid relative to the surface; and a controller to operate the set of actuators with a non-dimensional acceleration scale multiplier, a of at least 0.001 for fluid flows having friction Reynolds numbers, ReT, greater than 180, wherein the non-dimensional acceleration scale multiplier, a+, is a function of a non- dimensional actuation velocity, A+, divided by a non-dimensional actuation period, T+, where the non-dimensional velocity, A+, is equal to an amplitude of a spanwise actuation velocity, A, divided by a friction velocity, uT, of the fluid, such that A+= —andwhere the non-dimensional period, T+, is equal to an actuation period, T, multiplied by the square of a friction velocity,divided by a kinematic viscosity, v, of7’-.2 the fluid, such that T+= —12. The system of claim 11, wherein the non-dimensional period, I is at least 100.

13. The system of claim 11, wherein the non-dimensional period,is expressible as: where / is an actuation frequency of at least 0.1 Hertz.

14. The system of any one of claims 11-13, wherein the controller operates the set of actuators with a sinusoidal actuation displacement.

15. The system of any one of claims 11-13, wherein the controller operates the set of actuators with a non-sinusoidal actuation displacement.

16. The system of claim 15, wherein the controller operates the set of actuators with a non-sinusoidal waveform for actuation displacement to maximize cycle-integrated acceleration.

17. The system of any one of claims 11-13, wherein at least some of the set of actuators are spaced on the surface in a direction transverse to the flow direction of the fluid with a spacing distance between one-tenth and one-half of a boundary layer thickness, 8, of the fluid, where the boundary layer thickness, 8, of the fluid is calculated as a function of the fluid, the friction velocity, zr, of the fluid, and the kinematic viscosity, v, of the fluid.

18. The system of any one of claims 11-13, wherein the object is a pipe.

19. The system of claim 18, wherein each actuator comprises an oscillating surface actuator.

20. The system of claim 18, wherein the set of actuators includes at least one actuator to azimuthally oscillate a section of the pipe.

21. The system of claim 18, wherein the set of actuators includes a plurality of actuators to azimuthally oscillate a plurality of sections of the pipe.

22. The system of claim 18, wherein the pipe operates to implement at least one fluid operation selected from a group of fluid operations consisting of: transporting the fluid; mixing the fluid with one or more other substances; transferring heat from or to the fluid; and managing a chemical reaction of the fluid with a reactant.

23. The system of any one of claims 11-13, wherein the object is selected from a group of objects consisting of: a fixed-wing aircraft, a rotary-wing aircraft, a rocket, a missile, a projectile, a pump, a fan, a turbine, a wind turbine, a mast, an airfoil, a hydrofoil, a sail, a boat rudder, a boat hull, a rocket nozzle, a vehicle, and a duct.

24. The system of any one of claims 11-13, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of: air, water, and oil.

25. A method to modify drag, comprising: identifying fluid flow characteristics of a turbulent boundary layer of a fluid flowing relative to a surface of an object having friction Reynolds numbers, Re , greater than 180, the fluid flow characteristics including a friction velocity, ih, of the fluid and a kinematic viscosity, v, of the fluid; selecting a non-dimensional acceleration scale multiplier, a+, of at least 0.001 based on the identified fluid flow characteristics of the fluid flow; calculating, for the selected non-dimensional acceleration scale multiplier, a+, an actuation frequency, and actuation velocity, A, for injecting momentum along the surface of the object and perpendicular to a direction of fluid flow relative to the surface of the object, wherein the non-dimensional acceleration scale multiplier, a+, is expressible as a+= vf A / u^ andactuating a plurality of actuators on the surface of the object with the calculated actuation frequency, to selectively increase or decrease the drag of the fluid on the surface of the object.

26. The method of claim 25, wherein identifying the friction velocity, th, and the kinematic viscosity, v, of the fluid comprises estimating fluid flow characteristics based on one or more real-time measurements of the fluid flow.

27. The method of claim 25 or 26, wherein identifying the friction velocity, uT, and the kinematic viscosity, v, of the fluid comprises calculating fluid flow characteristics based on a velocity of the fluid relative to the surface.

28. The method of claim 25 or 26, wherein the actuation frequency is at least 0.1 Hertz.

29. The method of claim 25 or 26, wherein actuating the plurality of actuators operates to disrupt large-scale eddies in the turbulent boundary layer, where the large-scale eddies have a time scale that is at least 300 times larger than a viscous time scale, q’, where q’ is calculated as the identified kinematic viscosity, v, divided by the square of the identified friction velocity,30. The method of claim 25 or 26, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of: air, water, and oil.

31. The method of claim 25 or 26, further comprising identifying a friction Reynolds number, Re , of the fluid flowing relative to the surface of the object.

32. The method of claim 25 or 26 wherein the object is a pipe.

33. The method of claim 32, wherein each actuator comprises an oscillating surface actuator.

34. The method of claim 32, wherein the plurality of actuators includes at least one actuator to azimuthally oscillate a section of the pipe.

35. The method of claim 32, wherein the plurality of actuators azimuthally oscillates a plurality of sections of the pipe.

36. The method of claim 25 or 26, wherein the object is selected from a group of objects consisting of: a fixed-wing aircraft, a rotary-wing aircraft, a rocket, a missile, a projectile, a pump, a fan, a turbine, a wind turbine, a mast, an airfoil, a hydrofoil, a sail, a boat rudder, a boat hull, a rocket nozzle, a vehicle, and a duct.

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