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Control of fluid flow is an important, underutilized process possessing potential benefits ranging from avoidance of separation and stall on aircraft wings to reduction of friction in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier-Stokes (N.-S.) equations, whose solutions describe such flows, consist of a system of time-dependent, multidimensional, nonlinear partial differential equations (PDEs) which cannot be solved in real time using current computing hardware. The poor man's Navier-Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic—hence, easily (and rapidly) solved—and yet which retains many (possibly all) of the temporal behaviors of the PDE N.-S. system at specific spatial locations. Herein, we outline derivation of these equations and discuss their basic properties. We consider application of these equations to the control problem by adding a control force. We examine the range of behaviors that can be achieved by changing this control force and, in particular, consider controllability of this (nonlinear) system

Given the ever-increasing capabilities of computers and electronic technology, the ability to control a fluid with a range of actuators is already being investigated [

Control schemes may be open loop or closed loop, having different methods of actuation or correction and different algorithms to provide instruction and commands—largely depending on whether open or closed. At present, some methods include microelectromechanical systems (MEMSs) sensors and actuators (as in [

The notion of turbulence control has been researched for roughly the past 20 years, with a substantial mathematical formalism contributed by Abergel and Temam [

We propose herein the direct modeling of flow velocities through a discrete dynamical system (DDS) with the addition of an adjustable body—or control—force to achieve a desired system response. The behavior of this system has been shown to exhibit many—possibly all—of the temporal behaviors found in the full (PDE) N.-S. system at specific spatial locations. Through a straightforward process, the full N.-S. system is reduced to a coupled discrete dynamical system of equations resembling the much studied logistic map ([

Using a DDS to model naturally occurring phenomena is not a new idea—ecologist Robert May constructed the logistic map to model population dynamics [

To implement a control force in the system of PMNS equations, we add a body force to the right-hand side of the N.-S. equations. Since the PMNS equations model local velocities directly, a positive constant control force generally increases the value of the velocity in the respective direction. From studies such as [

Turbulent flows are known to have relatively high dimension for even low Reynolds number due to the spatial and temporal scales over which they occur (see [

In the following, sections we will outline the derivation of the PMNS equations beginning with the full PDE system of equations. The behavior of the DDS in general is the subject of continuous research, a precis of which will be included herein. We then present regime maps illustrating the types of behaviors that the PMNS equations are capable of reproducing and the corresponding bifurcation parameter and body force values for each regime. The aforementioned system complexity due to high codimension motivates the creation of the regime maps. In doing so, the system behavior is understood throughout the domain of bifurcation parameter space. From these regime maps, we deduce appropriate control forces and then implement these while iterating the PMNS equations to demonstrate controllability of the system toward a desired behavior. Time series of these results are shown.

Here, we describe the treatment applied to the N.-S. equations by which we derive the PMNS equations. After the derivation, we present a brief discussion of the features that are common to PMNS and full N.-S. equations, specifically the symmetry between equations and highly coupled nature of the full PDE system. Then we discuss the extension of these equations to a control context.

We begin with the incompressible N.-S. equations,

As is frequently used in theoretical analysis of N.-S. equations, we employ a Leray projection to a divergence-free subspace of solutions, thus eliminating the pressure gradient in (

Consistent with the mathematical understanding of the N.-S. equations (see, e.g., [

The Galerkin procedure is applied to (

Considering the equations without the imaginary factor (which can be removed

Removing all but a single arbitrary wavevector in (

Thus, the advanced time step equations can be expressed as

Now define the following bifurcation parameters:

For the study performed herein, we consider the isotropic case, which means that the

Note the similar structure of all three components of (

In order to carry out the intended computational experiments, we must begin with valid ranges for the bifurcation parameters

The following section contains results collected

Here, we present results from iteration of the DDS of (

The majority of the data presented in this work was calculated on a 376-node Dell high-performance computing cluster at the University of Kentucky Computing Center. Regime map calculations were performed in parallel using OpenMP with three cores from a 2.66 GHz Intel Xeon X5650 six-core processor. Less intensive calculations (

As in the studies [

Our goal is to distinguish the various behavior types which the DDS will produce, and do so automatically and objectively. Though some insight may be gained by calculating the fractal dimension, Lyapunov exponents, or other statistical quantities (or by constructing bifurcation diagrams), these will not allow automatic regime classification of the chaotic behaviors of which the DDS is capable. (Some comparison of these methods and their effectiveness is given in [

Identification of the regime type needs to be only qualitative. Thus, the PSD appears to contain sufficient information to permit distinguishing one regime from an inherently different one. The following is a list of the regime types which we have identified in the time series produced by the DDS

(a) 2501 × 1251 regime map for

Some discussion of the nomenclature and criteria for categorizing the regimes is due. Several of the regimes are referred to as “noisy” along with another description which identifies a distinguishing or primary characteristic of the behavior. This is to imply some broadband features of the PSD along with a salient distinguishing characteristic. Differing from laboratory experiments, the numerical DDS evaluation is of course done entirely within the computer. Thus, there is no “noise” in the sense of the type of signal fluctuation coming from instrumentation or sensors as would be a concern in laboratory experiments. The broadband noise which is observed in the PSDs we consider herein is an actual feature of the DDS. It has been shown in [

Figure

A key of the colors corresponding to each behavior type is presented in Figure

After the system bifurcates from subharmonic behavior, as the value of control force and bifurcation parameter are increased, there are several locations where the boundaries between the regimes are not well defined. This result is shared with the 2D and 3D studies of the PMNS equations in [

The same type of behavior is observed in Figures

The ordered sequence of bifurcations presented in Figure

To better exhibit the range of control possibilities within a small range of bifurcation parameter and control force, a localized regime map is presented in Figure

There is a tendency for regime types to orient themselves into regions aligned along the direction of varying control force (vertically in the regime maps). When attempting to vary the control force to achieve a different regime type, this presents a problem. In these regions, shown clearly in Figure

Despite this obstacle, there are still regions where the control force may be varied to achieve a particular regime with

In this subsection, we use contents of the regime map in Figure

Figure

Time series response of the PMNS equations. Control force

The regime map in Figure

The ability to control the

Time series response (

It is clear from both Figures

A noticeable difference between control of the PMNS system with streamwise control force (Figure

Significant research has been conducted in recent years to control dynamical systems (e.g., [

Though the PMNS dynamical system models the motions of fluids, the attempts to control this system may be compared with other studies in which a more general dynamical system is controlled. In [

Results from those general dynamical systems previously discussed have much in common with the time series results presented in Figures

Here, we present one (of probably many) possible algorithms by means of which the PMNS equations might be used to control turbulent fluid flow. It is important to observe that these equations can be rapidly evaluated, as already emphasized, and their simple algebraic structure is such as to permit easy implementations on microprocessors.

For simplicity, we assume that the desired flow regime is known, although one can easily envision situations where this might not be true. Then to achieve this desired behavior, carry out the following steps.

Collect flow data, possibly (but not necessarily) at many locations.

Use these data to construct PMNS equation bifurcation parameters.

Run the PMNS equations and use the regime map algorithm to identify the current physical flow regime.

If regime is the desired one, return to 1. If it is not, begin systematic perturbation of control force parameters with PMNS equation and regime map calculations until desired flow regime is obtained.

Convert PMNS equation control force to physical one and send corresponding signal to controller.

We remark that there are a number of details still to be treated in this control procedure, not the least of which is the need to perform implementations on microprocessors. But in this regard, the only part of the algorithm involving more than a few lines of code is the regime map program, and even this is relatively small and should easily fit on modern microprocessors. The more difficult parts of the algorithm are physically collecting sufficient data to build PMNS equation bifurcation parameters with adequate accuracy to be of use, and similarly inverting the process to convert PMNS equation control parameters to physical ones. Clearly, these issues are system dependent; some work is currently in progress to permit investigation of these ideas for a particularly simple system.

In this study, we have derived a discrete dynamical system directly from the 3D, incompressible Navier-Stokes equations and investigated the use of these equations within the context of turbulence control. We have studied the possible behaviors which the PMNS equations may exhibit and included computational results for

Derivation of the PMNS equations is, in principle, quite general and can be applied to a wide variety of problems governed by PDEs and (possibly) time-delay ODEs (e.g., models of machining processes). The derivation does not introduce any nonphysical quantities or attempt to model any physical ones. The PMNS equations have been shown to have significant potential as a “synthetic velocity” model, and herein, the ability to manipulate velocity fields across a wide variety flow behaviors was shown. The subject of ongoing investigations and future work will be implementation of the PMNS equations into hardware for use within a real-time control system similar to that described above.

3D velocity vector

Kinematic viscosity

Pressure

Reynolds number

Time

Spatial domain in

Wavevector

Normalization constant,

Time step parameter

Bifurcation parameter,

Bifurcation parameter,

Iteration counter