Spanwise wall forcing to modify a heat-transfer rate in a fluid flow

Abstract

A fluid control system includes at least one actuator on an object surface capable of generating an in-plane wave transverse to the fluid flow direction. A controller operates the actuator at an actuation frequency to increase or decrease a heat-transfer rate between the fluid and the surface. In some examples, the actuation frequency is chosen such that the ratio of the modified heat-transfer rate between the fluid and the surface to an un-actuated heat-transfer rate exceeds the ratio of the modified drag between the fluid and the surface to the un-actuated drag.

Description

Spanwise Wall Forcing to Modify a Heat-Transfer Rate in a Fluid FlowCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application Number 63 / 518,877 filed on August 10, 2023, titled “Simultaneous Drag and Heat Transfer Control By Spanwise Wall Forcing,” 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 formation and characteristics of the turbulent boundary layer also affect the heattransfer rate between the fluid and the surface of the object. In many industrial and commercial applications, controlling heat transfer between the fluid and the surface is as important or even more important than controlling the drag. In some instances, it may be desirable to control both the heat-transfer rate and the drag.

[0004] 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 describedherein 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.

[0005] The turbulent boundary layer impedes the motion of the object relative to the fluid around it and generally increases the heat transfer between the fluid and the object. 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. Turbulent flows in the turbulent boundary layer can be generally characterized as exhibiting fluctuations in pressure and flow velocity.

[0006] 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 100r|’ 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 300r|’.

[0007] 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).

[0008] 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 ( / «). The large-scale eddies and the small-scaleeddies 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 large-scale eddies and the small-scale eddies in the flow modify the heat-transfer rate between the object and the fluid. The Prandtl number of the fluid affects the heat-transfer rate and is defined as the ratio of the momentum diffusivity to the thermal diffusivity of the fluid. The presently described systems and methods include the use of transverse momentum injection to modify heat-transfer rates and / or drag between a fluid and a surface.

[0009] 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 valueand a fluctuating valueAS 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 q = v / U , where v is the fluid kinematic viscosity and uTis the friction velocity. This is often called the viscous length scale. The friction velocity, uT, can be expressed as uT= where p is the fluid density. The smallest eddies have a characteristic timescale q ’, which is defined as ' = — . Accordingly, the length scale, q, and the time scale, q’, ofthe smallest eddies can be characterized as:

[0012] The largest eddies in the turbulent boundary layer have a characteristic length scale, q0, equal to the boundary layer thickness, 3, and a characteristic time scale, qo' , equal to / EE. In 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,

[0013] 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:

[0014] Av = 3UT / V. Equation 3

[0015] 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, Re^ increases. By way of example, the friction Reynolds number, Re , associated with the flow of air over an airplanefuselage may be approximately 100,000, while the friction Reynolds number, 7?er, 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.

[0016] Attempts to reduce skin friction drag have focused on mitigating or modifying small- scale eddies in flows with friction Reynolds numbers, Re?, 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 Fa Commun. 12, 1-8 (2021) (hereinafter “Marusic”); 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.”

[0017] 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. Moreover, a turbulent Prandtl number, Prt, for a fluid is commonly used to describe or analyze heat transfer in turbulent boundary layer flows. Despite simple models for turbulent Prandtl numbers using a Reynolds analogy, the turbulent Prandtl number, Prt, for most fluids has been experimentally verified to generally not be equal to unity (1).

[0018] Much of the prior focus has been on controlling eddy formation to reduce drag. The existing literature does not suggest, and in some instances even teaches away from, disrupting eddy formation to modify heat-transfer rates independent from drag reduction. Moreover, existing literature does not suggest spanwise surface oscillations for transverse momentum injection to attain a modified heat-transfer rate that exceeds any achieved drag reduction.BRIEF DESCRIPTION OF THE DRAWINGS

[0019] FIG. 1A illustrates a performance plot of the fractional changes in the skin-friction coefficient and Stanton number for example surfaces, according to one embodiment.

[0020] FIG. IB illustrates a performance plot of the fractional changes in the skin-friction coefficient and Stanton number for other example surfaces, according to one embodiment.

[0021] FIG. 1C illustrates a performance plot of the fractional changes in the skin-friction coefficient and Stanton number for another example surface, according to one embodiment.

[0022] FIG. 2A illustrates examples of spanwise wall oscillations, according to one embodiment.

[0023] FIG. 2B illustrates a plot of data points for ratios of heat-transfer rate and drag reduction for spanwise plane oscillations of turbulent channel flows, according to one embodiment.

[0024] FIG. 3A illustrates a plot of the heat-transfer rate and drag reduction for various traveling wave frequencies for a fluid with a Prandtl number of 0.71, according to one embodiment.

[0025] FIG. 3B illustrates a plot of the heat-transfer rate and drag reduction for various traveling wave frequencies for a fluid with a Prandtl number of 4.0, according to one embodiment.

[0026] FIG. 3C illustrates a plot of the heat-transfer rate and drag reduction for various traveling wave frequencies for a fluid with a Prandtl number of 7.5, according to one embodiment.

[0027] FIG. 3D illustrates a plot of the heat-transfer rate and drag reduction for various plane oscillation frequencies for a fluid with a Prandtl number of 0.71, according to one embodiment.

[0028] FIG. 3E illustrates a plot of the heat-transfer rate and drag reduction for various plane oscillation frequencies for a fluid with a Prandtl number of 4.0, according to one embodiment.

[0029] FIG. 3F illustrates a plot of the heat-transfer rate and drag reduction for various plane oscillation frequencies for a fluid with a Prandtl number of 7.5, according to one embodiment.

[0030] FIG. 4A illustrates a plot of the heat-transfer rate reduction for fluids having various Prandtl numbers and Reynolds numbers, according to one embodiment.

[0031] FIG. 4B illustrates a plot of the difference between the reduction in the heat-transfer rate and the drag reduction for fluids having various Prandtl numbers and Reynolds numbers, according to one embodiment.DETAILED DESCRIPTION

[0032] 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.

[0033] 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, transfer heat from fluids, or manage chemical reactions of fluids. Examples of fluids for which inplane, transverse momentum injection may decrease (or selectively increase) friction and / or heat transfer include, but are not limited to, air, water, gaseous mixtures, natural gas, various liquids, oil, molten salt, and the like.

[0034] 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 inconcert 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. In addition, or collectively, a controller may control the actuation of any number of actuators, and multiple controllers and associated sets of actuators to operate in concert to achieve a target heat transfer profile, heat transfer reduction, or heat transfer 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.

[0035] For example, a controller may operate or actuate a plurality of actuators according to an actuation frequency, in Hz so that the period of the motion is given by T = !f and the angular frequency co is defined as a> = 2 f rad / s. In terms of the viscous time scale, 77', the period T can be expressed as a non-dimensional period T+(also referred to as a time scale multiplier) defined as T+For any given Rex, T}' is constant. When the controller is set to operate with a T+in the order of 100, the actuation period (T) is similar to the timescale of (and therefore actuates) the small-scale eddies in the flow (which have time scales on the order of lOOr ). Similarly, because large-scale eddies have time scales exceeding 300q’, when the controller is set to operate with a T+of more than 300, the actuation period (T) is similar to the timescale of (and therefore actuates) the large-scale eddies in the flow.

[0036] 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 the 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, the Prandtl number of the fluid, a mean or bulk velocity, U, a friction velocity, uT, of the fluid, a kinematic viscosity, v, of the fluid, and friction Reynolds numbers, ReT, of the fluid.

[0037] The controller may calculate an actuation frequency, f for transverse momentum injection along the surface (e.g., for in-plane momentum injection co-planar with the surface). Theactuator frequency, may be selected to disrupt large-scale eddies in the turbulent boundary layer. Again, the terms “large-scale” and “small-scale” are used as descriptors to distinguish between two types of eddies in a turbulent boundary layer.

[0038] The controller may actuate the plurality of actuators on the surface of the object using the calculated or determined actuation frequency, / to disrupt the large-scale eddies to selectively increase (or decrease) the drag of the fluid on the surface of the object and / or reduce (or increase) a heat-transfer rate of the fluid. The time scale multiplier, 7 / is a function of the identified friction velocity squared,divided by the product of (i) the calculated actuation frequency, / and (ii) the identified kinematic viscosity, v. Thus, the time scale multiplier, T is expressible as:

[0040] A fluid control system may include an actuator on the surface of an object to selectively generate an in-plane wave or plane oscillation to move the surface transverse to a flow direction of a fluid flow relative to the surface. The fluid flow may have a friction Reynolds number, ReT, greater than 180. The fluid may have a known or measured Prandtl value greater than, for example, 1.0.

[0041] A controller of the fluid control system may operate the actuator according to an actuation frequency and time scale multiplier to attain a modified heat-transfer rate between the fluid and the surface and / or a target drag reduction. The controller may operate an actuation frequency selected such that a ratio of the modified heat-transfer rate between the fluid and the surface to an un-actuated heat-transfer rate exceeds a ratio of a modified drag between the fluid and the surface to an unactuated drag. As described herein, the surface may be the surface of a pipe, a duct, 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, or the like. For example, the surface may be an internal or external surface of a pipe or other object. A pipe may be used to, for example, transport the fluid, mix the fluid with one or more other substances, transfer heat from or to the fluid, and / or manage a chemical reaction of the fluid with a reactant. The fluid may be, for example, air, water, oil, molten salt, or the like.

[0042] According to various embodiments, a fluid control system may include a controller that operates to identify fluid flow characteristics of a turbulent boundary layer of a fluid flowingrelative to the surface of an object. Examples of fluid flow characteristics include but are not limited to, a Prandtl number of the fluid, a friction velocity of the fluid and a kinematic viscosity of the fluid, a Reynolds number of the fluid flow, and / or the like.

[0043] The controller may select or operate according to a fixed or adjustable time scale multiplier (e g., a time scale multiplier greater than 50). The controller actuates one or more actuators to generate a surface wave or oscillate a surface based on a calculated actuation frequency. The controller may calculate the actuation frequency to inject momentum along the surface. The injected momentum is transverse to the direction of fluid flow. The frequency is selected to modify a heat-transfer rate between the fluid and the surface and is calculated as a function of the Prandtl number, the friction velocity, the time scale multiplier, the kinematic viscosity, and / or the Reynolds number. For Prandtl numbers exceeding approximately 1.0, the percentage change in the heat-transfer rate between the fluid and the surface may exceed any achieved percentage change in the drag reduction.

[0044] In various embodiments, a fluid control system includes various sensor devices to measure the characteristics of a fluid flow, a controller, and one or more actuators. An actuator may operate to cause a plane oscillation of the surface (e.g., a spanwise plane oscillation) that is transverse to a flow direction of the fluid. The fluid flow may, for example, have a friction Reynolds number greater than 180 with a fluid Prandtl value greater than 1.0. The controller may actuate the surface with an actuation frequency based on the Prandtl number, the Reynolds number, and a time scale multiplier, T of at least 50 to reduce a heat-transfer rate between the fluid and the surface and reduce drag between the fluid and the surface. In various embodiments, the actuation frequency may be selected such that the heat-transfer rate reduction exceeds the drag reduction on a percentage change basis.

[0045] 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) theyperform 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.

[0046] 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.

[0047] 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.

[0048] 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. Theright to add any described embodiment or feature to any one of the figures and / or as a new figure is explicitly reserved.

[0049] 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.

[0050] Various systems and methods are described herein for controlling the heat transfer between a fluid flow and a surface (e.g., a pipe surface or “wall” of an object). In some embodiments, the systems and methods also modify the drag of the fluid.

[0051] FIGS. 1A-C illustrates examples of approaches to modify or control both the heattransfer rate and drag of a fluid relative to a surface, according to various embodiments. The performance plots 110, 120, and 130 are used to measure or visualize the drag reduction and / or heat transfer efficacy of a particular surface modification approach. The performance of any given approach is commonly based on the changes achieved in the skin-friction coefficient, , and the Stanton number, Ch, as expressed in Equations 5 and 6 below,

[0052] =2 WEquation 5

[0054] In Equations 5 and 6, TW, qw, p, cp, and Twrepresent the wall shear-stress, wall heatflux, fluid density, specific heat capacity, and wall temperature, respectively. In a boundary layer, I7ref and Trefrepresent the free-stream velocity and temperature. In a channel flow, l / refand Trefrepresent the bulk velocity and either the bulk temperature or the mixed-mean temperature. The skin-friction coefficient, , and the Stanton number, C}l, are measured in the flow with surface control or surface modification, while the reference skin-friction coefficient,o, and the reference Stanton number, ChQ1are measured in the absence of control or absence of any surface modification at matched Reynolds and Prandtl numbers.

[0055] FIG. 1 A illustrates a performance plot 110 on a log-log scale of the fractional changes in the skin-friction coefficient, Rf, where Rf = / o, and the fractional changes in the Stanton number, Rh, where Rh= Ch / Ch0, for example surfaces 115 with grit and irregular roughness. The diagonal dashed line in the performance plot 110 represents the Reynolds analogy line where thefractional change in the skin-friction coefficient, Rf, (corresponding to the drag) is equal to the fractional change in heat transfer, Rh, (corresponding to the heat transfer). Data points deviating from the dashed line represent breaks from the Reynolds analogy.

[0056] Portions of the performance plot 110 within the Type A shading represent cases that demand heat-transfer augmentation with minimum pumping power and, hence, minimum drag. For example, the region of the performance plot 110 with Type A shading might be useful for cooling passages of gas turbine blades or solar air collectors. Portions of the performance plot 110 within the Type B shading represent cases that demand heat-transfer reduction with minimum pumping power. For example, the region of the performance plot 110 with Type B shading might be useful for crude oil transportation or heat energy transportation. In such applications, the fluid may be actively heated to reduce its viscosity and associated friction loss. Accordingly, it is desirable to reduce or minimize the heat transfer (e.g., heat loss) from the fluid to the surface (e.g., heat transferred from the fluid to the inner surface of a pipe).

[0057] The example surfaces 115 with grit and irregular roughness augment the heat transfer (Rh> 1), but for most data points, the drag increase exceeds the heat-transfer increase ( / ?y > Rh> 1), and so they fall outside the objective space for either Type A or B applications. For surface protrusions like roughness, viscous drag and pressure drag both contribute to the skin-friction coefficient, . However, the Stanton number, Ch, is associated with the wall heat-flux, which is the analog of the viscous drag. Therefore, the skin-friction coefficient, , has an additional positive component from the pressure drag, leading to fractional changes in the skin-friction coefficient, Rf, exceeding the fractional changes in the Stanton number, Rh, (Rf > Rh).

[0058] FIG. IB illustrates a performance plot 120 on a log-log scale of the fractional changes in the skin-friction coefficient, Rf, and the fractional changes in the Stanton number, Rh, for surfaces 125 with riblets. Again, the diagonal dashed line in the performance plot 120 represents the Reynolds analogy line. The two-dimensional streamwise-aligned surface protrusions do not induce significant pressure drag. Accordingly, most of the data points are scattered near the Reynolds analogy line, but a few data points break the analogy towards the regions for Type A or B applications.

[0059] FIG. 1C illustrates a performance plot 130 on a log-log scale of the fractional changes in the skin-friction coefficient, Rf, where Rf = / o, and the Stanton number, Rh, where Rh=Ch / Ch0, for surfaces 135 with blowing / suction. The diagonal dashed line in the performance plot 120 represents the Reynolds analogy line. The surface 135 with blowing / suction breaks the Reynolds analogy to a higher degree towards the objective space for Type A applications, yielding simultaneous heat-transfer augmentation and drag reduction (Rh> 1 > Rf >).

[0060] In the context of the systems and methods described herein, spanwise surface oscillations (e.g., spanwise wall oscillations) can be expressed as:

[0061] Ws(x, t) = A sin(Kxx + cot) Equation 7

[0062] In Equation 7, Wsis the instantaneous spanwise surface velocity that oscillates with amplitude, A, and frequency, a) = 2 rf . For non-zero values of the wavenumber, KX, the surface is actuated with a streamwise traveling wave. A simple spanwise oscillating plane is represented by Equation 7 with the wavenumber, KX, equal to zero (0).

[0063] FIG. 2A illustrates two examples of spanwise wall oscillations, including a first surface motion 210 with a non-zero wavenumber, KX. that generates a streamwise traveling wave on the surface, and a second surface motion 220 where the wavenumber, KX, is zero, resulting in a spanwise oscillating plane. With the optimal values of amplitude, A, frequency, m, and wavenumber, KX, wall oscillations can reduce drag by 40%.

[0064] FIG. 2B illustrates a plot 230 of data points for streamwise traveling wave oscillations (i.e., where KX^0, square symbols), and for spanwise plane oscillations (i.e., where KX= 0, circle symbols) of turbulent channel flows at a Reynolds number, ReT, of 590 and various Prandtl numbers. The turbulent channel flow is characterized by ReTo= huTo / v, where h is the channel half-height, UTQ= JTWQ / P, and v is the fluid kinematic viscosity.

[0065] By operating at a large viscous-scaled actuation period T+sc= 2n / <ji)+= 2nuj:o / (coy) = 500 and a large amplitude of oscillation A+= A / UTQ= 30, the drag increased by 7.7% (i.e., Rf = 1.077), while the heat transfer increased by 15% (i.e., Rh= 1.15), such that this actuation paradigm falls into the objective space for Type A applications.

[0066] As understood by those of skill in the art, the superscript “+” indicates normalization using the viscous length V / UTQand velocity UTQscales. The superscript indicates viscous scaling based on the actual friction velocity, that is, uT, of the drag-altered flow for the actuated cases. The drag for turbulent forced convection in a channel flow is expressible as:

[0068] The heat transfer includes the Prandtl number, Pr, where Pr = v / a, for turbulent forced convection in a channel flow is expressible as:

[0070] The drag reduction is given by DR — 1 — Rf, and the heat-transfer reduction is given by HR = 1 — Rh. The plot 230 includes data points for fluids having various Prandtl numbers, including air (Pr = 0.71), water (Pr = 7.5), and molten salt (Pr = 20.0). We vary Pr to assess the performance of (1.2) in different fluids, including air (Pr = 0.71), water (Pr = 7.5), and molten salt (Pr = 20.0). The data points are from a fixed amplitude of oscillation, A+, of 12 for a spanwise oscillating plane (KX= 0) and a traveling wave (KX= 0.0014) to produce a traveling wave with + — 4500, indicated respectively with filled circles and squares. The frequency, co+, is varied from 0.022 to 0.110, corresponding to a regime that provides significant drag reduction. The corresponding actuation parameters fall within the inner-scaled actuation pathway Tsc< 350 -> <jj+> 0.018, where the drag reduction, DR, and the heat-transfer reduction, HR, are achieved through attenuation of the near-wall turbulence in the inner region.

[0071] Notably, the data points in the plot 230 correspond to a decrease in both drag and heat transfer, with a maximum drag reduction of 30% and a heat-transfer rate reduction of 40%. For Prandtl numbers greater than one, however, the heat-transfer reduction is greater than the drag reduction, breaking with the Reynolds analogy and falling within the space for Type B applications.

[0072] FIGS. 3A-3F illustrate plots of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for various frequencies.

[0073] FIG. 3 A illustrates a plot 310 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for an upstream traveling wave on a surface with frequencies, m+, between 0 and 0.12 for a fluid with a Prandtl number of 0.71.

[0074] FIG. 3B illustrates a plot 320 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for a n upstream traveling wave on a surface with frequencies, a>+, between 0 and 0.12 for a fluid with a Prandtl number of 4.0.

[0075] FIG. 3C illustrates a plot 330 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for an upstream traveling wave on a surface with frequencies, a>+, between 0 and 0. 12 for a fluid with a Prandtl number of 7.5.

[0076] FIG. 3D illustrates a plot 340 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for a plane oscillation of a surface with frequencies, m+, between 0 and 0.12 for a fluid with a Prandtl number of 0.71.

[0077] FIG. 3E illustrates a plot 350 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for a plane oscillation of a surface with frequencies, m+, between 0 and 0.12 for a fluid with a Prandtl number of 4.0.

[0078] FIG. 3F illustrates a plot 360 of the drag reduction, heat-transfer reduction, and the difference between the heat-transfer reduction and drag reduction for a plane oscillation of a surface with frequencies, a)+, between 0 and 0.12 for a fluid with a Prandtl number of 7.5.

[0079] As demonstrated in FIGS. 3A-F, the heat-transfer rate and drag reduction initially increase with the frequency, m+, but for values greater than about 0.04, the heat-transfer rate and drag reduction reach a maximum value before slowly decreasing. The maximum drag reduction is about 30% for all Prandtl numbers, but the maximum heat-transfer reduction increases from 30% to about 40% with increasing Prandtl numbers, marking a significant break from the Reynolds analogy. The estimated optimum frequency for drag rate reduction is independent of the Prandtl number, at about 0.046. In contrast, and breaking with the Reynolds analogy, the optimum frequency for heat-transfer rate reduction depends on the Prandtl number of the fluid.

[0080] According to various embodiments, a fluid control system may cause an actuator to generate a traveling wave on a surface with an actuation frequency with oo+« 0.044 to maximize the drag reduction. In other embodiments, a fluid control system may cause an actuator to generate a plane oscillation of the surface with an actuation frequency with co+« 0.066 to maximize the drag reduction. For fluids with a Prandtl number of 0.71, these same actuation frequencies may also maximize (or approximately maximize) the heat-transfer rate as well. However, for Prandtl numbers above 0.71 (e.g., at 4.0 or 7.5), a higher action frequency may be used to increase the heat-transfer rate reduction. Such higher actuation frequencies may result in significantly higher heat-transfer rate reductions than drag reductions.

[0081] For example, to maximize the heat-transfer rate reduction in a fluid with a Prandtl number of 7.5, the fluid control system may actuate an actuator with an actuation frequency between 0.055 and 0.088. The fluid control system may actuate an actuator with an actuation frequency of 0.046 to maximize the heat-transfer rate reduction of a fluid with a Prandtl number of 0.71.

[0082] In various embodiments, a controller may actuate an actuator to generate a traveling wave on a surface or oscillate a surface to maximize a heat-transfer rate or maximize a reduction in the heat-transfer rate. In other embodiments, a controller may actuate an actuator to generate a traveling wave on a surface or oscillate a surface to maximize a difference between the reduction in the heat-transfer rate and the reduction in the drag rate.

[0083] FIG. 4A illustrates a plot 410 of the heat-transfer rate reduction (curves labeled 25 to 50) for fluids having various Prandtl numbers (horizontal axis) and Reynolds numbers (vertical axis). The plotted values correspond to a fluid control system that generates a traveling wave on a surface with an amplitude of oscillation, A+, of 12, a wavenumber, KX, of 0.0014, and an actuation frequency, co+, of 0.088. The bolded heat-transfer rate reduction values 28%, 37%, 40%, and 43% correspond to a fluid flow with a Reynolds number of 590 for Prandtl values of 0.71, 4.0, 7.5, and 20, respectively. As evidenced in FIG. 4A, at a given Reynolds number, the reduction in heattransfer rate increases almost logarithmically with the Prandtl number. In addition, the reduction in the heat-transfer rate of fluid with a given Prandtl number decreases very slowly with increasing Reynolds numbers. The values in plot 410 for a traveling wave on a surface are broadly valid for plane oscillation of the surface with only small variations in the specific values.

[0084] FIG. 4B illustrates a plot 420 of the difference (curves labeled -2 to 22) between the reduction in the heat-transfer rate and the drag reduction for fluids having various Prandtl numbers (horizontal axis) and Reynolds numbers (vertical axis). Again, the plotted values correspond to a fluid control system that generates a traveling wave on a surface with an amplitude of oscillation, A+, of 12, a wavenumber, KX, of 0.0014, and an actuation frequency, co+, of 0.088. The bolded differences of -1.5, 8.4, 10.6, and 13 correspond to a fluid flow with a Reynolds number of 590 for Prandtl values of 0.71, 4.0, 7.5, and 20, respectively.

[0085] As demonstrated herein, for fluid flows having Prandtl numbers greater than approximately 1.0, actuation frequencies above approximately 0.044 provide a reduction in heat-rate transfer that exceeds the drag reduction. In fact, the difference between the reduction in the heat-transfer rate and the drag reduction increases from approximately 5% for a Prandtl number of 4.0 to more than 10% for a Prandtl number of 7.5.

[0086] For plane oscillation of a surface, the optimal actuation frequency, co+, for the maximum reduction in heat-transfer rate is nearly the same as the optimal actuation frequency, co+, for the maximum drag reduction (e.g., oo+= 0.066). However, for traveling wave actuation, the optimal actuation frequency, co+, to maximize the reduction in heat-transfer rate changes from 0.044 for a fluid having a Prandtl number of 0.71 to 0.088 for a fluid having a Prandtl number of 7.5. According to various embodiments, a controller of a fluid control system utilizes a predictive model for the heat-transfer rate, in terms of the Prandtl number and the Reynolds number, to actuate one or more actuators to achieve a target reduction in heat-transfer rate, drag reduction, and / or a difference between the reduction in the heat-transfer rate and the drag reduction. In various embodiments, some of the actuation parameters may be fixed, while others may be varied by the controller. For example, the controller may vary an actuation frequency to achieve a target reduction in heat-transfer rate given a known Prandtl number of the fluid, a known (measured or provided) Reynolds number of the fluid flow, and fixed actuation parameters for the amplitude of oscillation, A+, and / or wavenumber, KX, where KX= 0 in the case of plane oscillation.

[0087] 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.

[0088] 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 beconstrued 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. A fluid control system, comprising: an actuator on a surface of an object that is selectively controllable to generate an inplane wave in the surface that is transverse to a flow direction of a fluid flow relative to the surface having friction Reynolds number, ReT, greater than 180 and a fluid Prandtl value greater than 1.0; and a controller to operate the actuator according to an actuation frequency, based on a time scale multiplier, T , of at least 50, to attain a modified heat-transfer rate between the fluid and the surface, wherein the time scale multiplier, T and the actuation frequency, are related by a function of a kinematic viscosity, v, of the fluid divided by the square of a friction velocity, u , of the fluid, and wherein the actuation frequency, / is selected such that a ratio of the modified heattransfer rate between the fluid and the surface to an un-actuated heat-transfer rate exceeds a ratio of a modified drag between the fluid and the surface to an un-actuated drag.

2. The system of claim 1, wherein the wavenumber of the in-plane transverse wave in the surface is zero (0), such that the actuator causes a spanwise plane oscillation of the surface.

3. The system of claim 1, wherein the time scale multiplier, , is at least 50.

4. The system of claim 1, wherein the time scale multiplier, 7 / is expressible as: actuation frequency.

5. The system of any one of claims 1-4, wherein the object comprises a pipe and the surface comprises an internal surface of the pipe or an external surface of the pipe.

6. The system of any one of claims 1-4, wherein the object is a pipe, and 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.

7. The system of any one of claims 1-4, wherein the object comprises a duct and the surface comprises an internal surface of the duct or an external surface of the duct.

8. The system of any one of claims 1-4, 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.

9. The system of any one of claims 1 -4, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of: air, water, molten salt, and oil.

10. 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, ReT, greater than 180, the fluid flow characteristics including a Prandtl number of the fluid, a friction velocity, U , of the fluid and a kinematic viscosity, v, of the fluid; selecting a time scale multiplier T+greater than 50; calculating an actuation frequency, for injecting momentum along the surface and transverse to a direction of fluid flow relative to the surface to modify a heat-transfer rate between the fluid and the surface, where the actuation frequency, is calculated as a function of the Prandtl number, the friction velocity, wr, the selected time scale multiplier, 7 / and the kinematic viscosity, v; and actuating one or more actuators on the surface at the actuation frequency, / to disrupt large-scale eddies in the turbulent boundary layer to reduce a heat-transfer rate between the fluid and the surface that exceeds a drag reduction for Prandtl numbers exceeding 1.0.11 . The method of claim 10, wherein the large-scale eddies have a time scale that is at least 300 times larger than a viscous time scale, r , where r is calculated as the identified kinematic viscosity, v, divided by the square of the identified friction velocity, u .

12. The method of claim 10, wherein calculating the actuation frequency, comprises calculating the actuation frequency, / such that a ratio of the modified heat-transfer rate between the fluid and the surface to an un-actuated heat-transfer rate exceeds a ratio of a modified drag between the fluid and the surface to an un-actuated drag.

13. The method of any one of claims 10-12, wherein identifying the friction velocity, Mr, 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.

14. The method of any one of claims 10-12, wherein identifying the friction velocity, Mr, and the kinematic viscosity, v, of the fluid comprises calculating fluid flow characteristics based on a velocity of the fluid relative to the surface.

15. The method of any one of claims 10-12, further comprising identifying a friction Reynolds number, ReT, of the fluid flowing relative to the surface of the object.

16. The method of any one of claims 10-12, wherein the actuation frequency, / is less than 20,000 Hz for friction Reynolds numbers, kv, greater than 180.

17. The method of any one of claims 10-12, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of: air, water, molten salt, and oil.

18. The method of any one of claims 10-12, wherein the object is selected from a group of objects consisting of: a pipe, 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.

19. A fluid control system, comprising: an actuator on a surface of an object to cause a plane oscillation of the surface that is transverse to a flow direction of a fluid flow relative to the surface having friction Reynolds number, ReT, greater than 180 and a fluid Prandtl value greater than 1.0; and a controller to operate the actuator according to an actuation frequency, based on a time scale multiplier, 7 / of at least 50, to reduce a heat-transfer rate between the fluid and the surface and reduce drag between the fluid and the surface; wherein the time scale multiplier, 7*, and the actuation frequency, are related by a function of a kinematic viscosity, v, of the fluid divided by the square of a friction velocity, u^, of the fluid, and wherein the actuation frequency, / is selected such that the heat-transfer rate reduction exceeds the drag reduction on a percentage change basis.

20. The system of claim 19, wherein the time scale multiplier,is at least 50.

21. The system of claim 19, wherein the time scale multiplier, 7 / is expressible as: actuation frequency.

22. The system of any one of claims 19-21, wherein the object comprises a pipe and the surface comprises an internal surface of the pipe or an external surface of the pipe.

23. The system of any one of claims 19-21, wherein the object is a pipe, and 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.

24. The system of any one of claims 19-21 , wherein the object comprises a duct and the surface comprises an internal surface of the duct or an external surface of the duct.

25. The system of any one of claims 19-21, 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.

26. The system of any one of claims 19-21, wherein the fluid comprises at least one fluid selected from a group of fluids consisting of air, water, molten salt, and oil.

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