Turbulence and Unsteady Fluid Mechanics Research:

The consistent themes in these research projects are the role of unsteadiness and the extent to which we can develop predictive models of turbulent motion in unsteady flows (as if it were not hard enough to do so in steady flows!).

Unsteady Laminar Flow Metering:
We have recently demonstrated a time-accurate flow rate sensor for unsteady laminar liquid flows of Newtonian fluids in pipes. The principle on which this sensor is based is a solution we have developed to the unsteady, fully-developed Navier Stokes equations, that expresses the momentary flow rate as a functional of pressure drop. The functional can be expressed as a term that is a multiple of the momentary pressure drop (a Hagen-Poiseuille term) and a convolution integral of pressure gradient history (an unsteady correction term). The unsteady correction term, which is analytically exact, is necessary because, in unsteady flows, the pressure drop is out of phase with the flow rate. This unsteady flow meter comprises two micro-pressure sensors imbedded in a long section of pipe, to measure the momentary pressure drop along the pipe, and signal processing equipment to construct the convolution integral and deduce the momentary flow rate. In test experiments, the principle appears to work perfectly for duct flows with arbitrary unsteadiness histories. Dr. Brereton holds a patent on this unsteady flow-metering technique.

Unsteady Turbulent Flow Metering:
In the unsteady flow studies described above, an exact solution is possible because the unsteady, laminar fully-developed-flow equations are linear and forward and inverse Laplace transformation, which facilitates the convolution representation, is possible. In the corresponding turbulent flow problem, it would be extremely useful to have an approximation to an unsteady correction factor. However, if the same approach is to be taken as for laminar flow, one must model the Reynolds stress in a closure which is at most a linear function of velocity, velocity gradient or y, possibly with an intrinsic time delay. We are presently evaluating the efficacy of such an approach through comparisons with experimental measurements of momentary flow rate and pressure gradient.

Unsteady Compressible Flow Metering:
One of the most challenging problems in automotive engine control is accurate metering of air in intake flows, which are highly unsteady. Today's automobiles typically use a hot-wire anemometer as the flow velocity sensor in a mass airflow sensor. For engine control purposes, it is desirable to read a single, steady voltage signal from the sensor that represents the integrated contribution of a highly unsteady flow over each engine cycle. The use of an instantaneous sensor without directional sensitivity in a bi-directional flow complicates matters further. When deployed in nearly all today's automobile engines, a highly damped voltage output from a potentially time accurate hot-wire sensor has to be calibrated in situ by making a look-up table to provide the correct measured average flow rate from voltage at fixed engine speeds. No calibration is made for the infinite number of possible transient histories and so engine control modules are typically programmed to disregard the sensor output during transients. So much for making good use of a potentially valuable and accurate sensor! We are presently exploring approaches to deducing the momentary flow rate in these transient, compressible, almost frictionless duct flows by using simple sensors, approximate solutions to the momentary equations of fluid motion, and smart (real-time analog electronics) algorithms.

Unsteady Laminar Flow Control:
In the area of flow control, we have used exact solutions to the unsteady Navier Stokes equations in fully-developed duct flows to find the optimal transients in flow rate that will accelerate fluid from one value to another with the least power input. In these problems, the power input is again a functional of flow rate and the optimal solution departs significantly from the quasi-steady one as less time is permitted for completion of the transient. We are presently examining DNS databases to see if similar functional considerations apply to optimization of turbulent flows.

Unsteady Turbulence Modeling:
The most difficult of the various areas of unsteady fluid mechanics we study is that of developing turbulence models for use in unsteady turbulent flows. While it is relatively easy to point to shortcomings of today's quasi-equilibrium two-equation models in unsteady flows, it is another matter entirely to propose better, more general models. We have explored some possibilities of using RDT (rapid distortion theory) results as asymptotic limits towards which turbulence models should adjust at high rates of strain, but real flows seem to reach only momentarily the conditions under which RDT assumptions would apply. Moreover, we are presently able only to compute RDT results in homogeneous turbulence, using the Reynolds-Kassinos formulation, for arbitrary strain-rate histories. It seems as though non-local modeling in both time and space may be necessary to provide adequate closures for Reynolds stress (and structural) descriptions of turbulence and research in this general area is of continued, long-term interest.

Selected Publications:

Brereton, G. J. & Mankbadi, R. R.
Development of a rapid-distortion turbulence model for unsteady hydrodynamic and scalar boundary layers.
Tenth Symposium on Turbulent Shear Flows, Penn State Univ., (1995).

Brereton, G. J.,
The interdependence of friction, pressure gradient, and flow rate in unsteady laminar parallel flow.
Phys. Fluids, 12, 3, 518, 2000.

Brereton, G. J. & Hwang, J-L.
The spacing of streaks in unsteady turbulent wall-bounded flow.
Phys. Fluids A, 6 (7), 2446--2454, (1994).

Brereton, G. J. & Reynolds, W. C.
Dynamic response of boundary-layer turbulence to oscillatory shear.
Phys. Fluids A, 3 (1), 178--187, (1991).

Brereton, G. J.
Stochastic estimation as a statistical tool for approximating turbulent conditional averages.
Phys. Fluids A, 4 (9), 2046--2054, (1992).

Brereton, G. J. & Kodal, A.
An adaptive turbulence filter for decomposition of organized turbulent flows.
Phys. Fluids 6 (5), 1775--1786, (1994).

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