ECE 533 Final Project Decomposing non-stationary turbulent velocity in open channel flow Ying-Tien Lin 2005.12.202Decomposing non-stationary turbulent velocity in open channel flow Ying-Tien Lin 1 Introduction In natural environment, the flow of a fluid can be categorized as laminar or turbulent flow. Observing that water run through a pipe, we can inject neutral dye to investigate the flow characteristics (see the Figure 1 below). For “small enough flowrates”, the dye streak will remain as a well-defined line as it flows along, with only slight blurring due to molecular diffusion of the dye into the surrounding water. For a somewhat “intermediate flowrates”, the dye streak fluctuates in time and space, and intermittent bursts of irregular behaviors appear along the streak. On the other hand, for ”large enough flowrates” the dye streak almost immediately becomes blurred and spread across the entire pipe in a random fashion. These three characteristics are denoted as laminar, transitional, and turbulent flow, respectively. Figure 1 Experiment to illustrate type flow and dye streaks3Suppose that we place a velocimetry probe at one point of the pipe, for laminar flow, there is only one component for velocity; however, for turbulent flow the predominant component of velocity is also along the pipe, but it is accompanied by random components normal to the pipe axis. Slow motion pictures of the flow can more clearly reveal the irregular, random, turbulent nature of the flow. Figure 2 Time independent of fluid velocity at a point Turbulent flows are beneficial to our daily life. Mixing processes and heat and mass transfer processes are considerably enhanced in turbulent flow compared to laminar flow. For example, to transfer the required heat between a solid and an adjacent fluid (such as in the cooling coils of an air conditioner or a boiler of a power plant) would require an enormously large heat exchanger if the flow were laminar. Furthermore, it is considerably easier to mix cream into a cup of coffee (turbulent flow) than to thoroughly mix two colors of a viscous paint (laminar flow). In order to deal with turbulent flow, previous researcher represented the turbulent velocity as the sum of time mean value, u and the fluctuating part of the velocity, 'u, that is: 'uuu=+ (1) Where 001()tTtuutdtT+=∫, the time interval, T, is considerable longer than the period of the longest fluctuations, but considerably shorter than any unsteadiness of the4average velocity. It is straightforward to prove that the time average of the fluctuation velocity, 'u is zero. The products of the fluctuation velocity in x and y components will account for the momentum transfer (hence, the shear stress). The total shear stressτ can be shown as follows: ''lam turbduuvdyτμρ ττ=− =+ (2) Where μ is the viscosity of the fluid, ρ is the density of the fluid. The customary random molecule-motion-induced laminar shear stress lamτ is /du dyμ. For turbulent flow it is found that the turbulent shear stress, ''turbuvτρ=− is positive. Hence, the shear stress is greater in turbulent than in laminar flow. Terms of the form ''uvρ− are called Reynolds stresses in honor of Osborne Reynolds who first discussed them in 1895. In a very narrow region near the wall, the laminar shear stress is dominant. Away from the wall the turbulent portion of the shear stress is dominant. In natural rivers, the shear stress is most relevant to the sediment transport, which means that it affects the amount of sediment brought from upstream to downstream (see Figure 3 below). Figure 3 Sediment transport in natural river As mentioned above, for extracting the time mean velocity, u, the unsteadiness of the average velocity is supposed to be very small. Suppose the turbulent flow follows a stationary process, the mean velocity u can be obtained easily by taking5the average of the instantaneous velocity u, that is, u is not a function of time, . Then, the fluctuation velocity 'u can be conventionally calculated by subtracting the mean velocity u from the instantaneous velocities u in the realizations of the flow. Hence, the total flow can be split into the mean part and the fluctuating component. However, under non-stationary flow field, the time-varying mean velocity failed to be obtained as easily as that in stationary flow. How to extract the time-varying mean velocity from the velocity records has become the initial and critical step to understand the turbulence characteristics of the non-stationary flow. The most common non-stationary turbulent flow occurs in the flooding period, where velocity profile changes with time rapidly. Due to the features in flooding period, taking average velocities as the time mean average values is unsuitable any more; hence, other methods are applied to decompose the instantaneous velocity profile. Refer to previous investigations in fluid mechanics (Song and Graf (1996), Nezu et al. (1997), Haung et al. (1998)) and some digital signal processing literatures (Jansen (2001), and Rioul et al.(1991)) (the velocity in turbulent flow resembles the contaminated signals in digital signal processing, my task is to separate the time mean value and fluctuation velocity, somewhat similar to the de-noising processes.), three methods will be applied in this project, which are Fourier decomposition method, wavelet transformation, and Empirical mode decomposition, respectively. The introductions of these methods are provided in the next section: 2 Decomposition methods Three methods used to decompose the contaminated signals are briefly introduced as follows: (1) Fourier decomposition method:6The Fourier component method is to transform the instantaneous velocity profile{}Niiu1= into the frequency domain by using the discrete Fourier transform (DFT). Only the frequency components lower than Tmf 2/)1(−= are used to represent the mean velocity component; m is an odd integer, T is the time period of the measurement of a hydrograph. The Fourier sum can be interpreted as the mean velocity component. ()∑++=−=2/)1(10sincos21miikkikkibaaUωω (3) Where ∑−==10cos2NiikikuNaω, ∑−==10sin2NiikikuNbω kNiik)/(2πω= ()2/)1(,...,2,1,0−= mk This is like to pass the signal through a low pass filter with cut-off frequency of Tmf 2/)1(−= . The frequency higher than the cut-off
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