A method for prediction of the spectrum observed with the laser Doppler flowmeter (LDF) is developed. A detailed analysis of the spectrum observed with steady flows is presented. The effects of diffusive motion of the scattering centers, spatially varying velocities, finite transit time of the centers through the sample volume, and uncertainty in scattering vector are computed. The latter two effects are shown to be identical. A computation of the sample volume size and shape is made using the optical parameters of the system. The calculated linewidths and shapes are in good agreement with data taken on two independent LDF systems. The general formalism is also applicable to measurements made on turbulent fluids and diffusely reflecting solid objects. The special case of a static fluid (zero velocity) is considered in detail and the low angle correction to the conventional expressions used in laser ``beat‐frequency'' spectroscopy is developed.

Spectral Analysis of the Signal from the Laser Doppler Flowmeter: Time‐Independent Systems

In this paper, two time relaxation parameters are introduce to a thermal lattice BGK, model to make its Prandtl number controllable. The dependency of the Prandtl number on the two parameters is derived. Numerical measurement of the transport coefficients is used to demonstrate the validity of the method. Furthermore, two examples of convective heat transfer are calculated, with one to show the effectiveness, and the other to show the breakdown of the two-parameter formulation under different conditions.

Two-Parameter Thermal Lattice BGK Model with a Controllable Prandtl Number

A new model of velocity distribution of a single-phase fluid flowing in packed beds

Micro-molecular tagging velocimetry (μMTV) has been used to characterize the hydrodynamic developing flow in a microtube inlet with a nominal inner diameter of 180 μm. Velocity profile data at 11 axial locations within the hydrodynamic developing region were acquired using the μMTV approach and the results represent the first characterization of hydrodynamically developing pipe flow at the microscale. The uncertainty in measurements of time-averaged velocity profiles ranged from 6% to 7% of the centerline velocity. The uncertainty in instantaneous measurements is in the range 8%-16% of the peak maximum velocity. Data were taken at Reynolds numbers of 60, 100, 140, 290, and 350. The data suggest the formation of a vena contracta with either locally turbulent flower unsteady laminar flow separation early in the tube for the larger Reynolds (Re) numbers, which is quite different from macroscale experiment or numerical simulation where a vena-contracta is not observed for Re < 500. The velocity profiles obtained very near the tube entrance exhibited a uniform velocity core flow surrounded by regions of relatively stagnant fluid in the near wall regions

Characterization of the Hydrodynamically Developing Flow in a Microtube Using MTV

A discrepancy exists between the predictions of analytical solutions of approximate Navier–Stokes equations and numerical finite-difference solutions of the full Navier–Stokes equations regarding the development of laminar flow at the entrance to cylindrical pipes for Newtonian fluids Starting from a uniform velocity profile at the entrance to the pipe, analytical solutions of approximate Navier–Stokes equations predict the velocity profile to have a maximum at the centre of the pipe at all times In contrast, numerical finite-difference solutions of the full Navier–Stokes equations have suggested that the location of the velocity maximum moves from the wall towards the centre of the pipe at a short distance from the entrance, after which it remains at the centre of the pipe This study presents the first experimental evidence of the moving velocity maximum from the wall towards the centre of the pipe The initial uniform velocity profile was achieved by flowing the fluid through a monolith composed of narrow parallel channels and the flow development was investigated using magnetic resonance velocimetry The experimentally observed variation of the position and size of the velocity maximum with the Reynolds number and the distance from the entrance to the pipe is shown to be in good agreement with the predictions of numerical finite-difference solutions of the full Navier–Stokes equations

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Experimental evidence of velocity profile inversion in developing laminar flow using magnetic resonance velocimetry

A laser doppler flowmeter is used to obtain velocity profiles in circular glass tubes downstream from a plug composed of brass screens. The apparatus for measuring point velocities of particles suspended in a fluid is described. Measurements were made very close to the plug at Reynolds numbers of 16.5, 47, and 274, and the flow development followed down the tube. The results show that the entrance flow development is dependent on the inlet profile shape and the Reynolds number for Reynolds numbers below 300.

The flow development at Reynolds numbers less than 300 was considerably slower than that predicted by the boundary-layer solution of the equations of motion starting with a flat profile.

Valerio A. Santos

Papers

Laminar velocity profiles in developing flows using a laser doppler technique