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Preliminary Concepts
Published in Hillel Rubin, Joseph Atkinson, Environmental Fluid Mechanics, 2001
Estimates for vl(j) may be obtained by direct measurements of the Reynolds stresses and mean velocity gradients, using Eq. (5.5.11). However, this is normally a difficult and time-consuming procedure, requiring a large number of measurements in order to obtain reasonable averages. Other estimates of turbulent viscosities may be made by conducting experiments where the diffusion of a conservative tracer is observed over time and values for vt(j) are chosen so that the equation fits the observations (assuming that at least the mean flow is well known). This procedure requires using a solution to the turbulent advection-diffusion equation for dissolved mass, which is described in Chap. 10, along with the application of the Reynolds analogy, which is an assumption that the turbulent diffusivities for momentum, mass, and heat are the same, since they all depend on the same eddies for transport. Turbulent viscosities also may be estimated as the product of a typical turbulent velocity scale, such as the root-mean-square velocity fluctuation, u'2-1/2, and an appropriate length scale, usually the integral length scale (analogous to the integral time scale defined in Eq. 5.2.9). This approach is similar to the mixing length approach used in defining molecular diffusivities, described in Sec. 10.3.1.
Pressure field in skimming flow over a stepped spillway
Published in H.-E. Minor, W.H. Hager, Hydraulics of Stepped Spillways, 2020
M. Sánchez Juny, J. Pomares, J. Dolz
where Re is the correlation coefficient between two points x and x+χ, with the same time origin. Therefore, the integral length scale could be conceived as a measure of the large eddies in the turbulent flow.
Flow and Thermal Field Measurement Techniques
Published in Je-Chin Han, Lesley M. Wright, Experimental Methods in Heat Transfer and Fluid Mechanics, 2020
Turbulence Length Scale Measurement: The integral length scale is representative of the size of the most energetic eddies in the flow. It can be determined through analysis of the autocorrelation of the fluctuating velocity time signal from hot wire anemometry, as shown in the following equations. Figure 11.27 shows the calculated turbulence length scale profile inside the boundary-layer, it increases from near wall (around 0.7 cm) to outside the boundary-layer (around 2.0 cm). () Autocorrelation of u′iRuu(t)=(u′(t)⋅u′(t+τ)¯uRMS2)=1N(∑1Nu′i⋅u′i+juRMS2) () T=∑1NoRuui⋅Δτ () τ=jΔt () Turbulent length scaleΛX=U¯⋅T
Experimental investigation of the influence of Reynolds number and buoyancy on the flow development of a plane jet in the transitional regime
Published in Journal of Turbulence, 2021
P. R. Suresh, T. Sundararajan, K. Srinivasan, Sarit K. Das
The integral time scale is calculated using autocorrelation and length scale is estimated for various Reynolds numbers (Figure 11). The integral length scale obtained represents the typical size of energy containing turbulent eddies. The figure shows an increase in length scale over the range of x/d for all Reynolds numbers. Although the value increases with distance, there is no linearity especially in the far region unlike the isothermal cases [9]. It is clear that at a given axial location, the length scale reduced with the increase of Reynolds number. It shows that turbulent scales become finer at higher Re values. A broad range of eddy sizes will coexist at any location. The integral length scales are larger for low Re jets. This can be attributed to the shear layer instability and the generated large vortices. It is clear that heating increases the size of the scales. This can be attributed to the expansion of the fluid. Hence it is clear that both the half width spread and length scales evolve in the same way. Increases over axial distance decrease with Reynolds number. The length scales can be represented as The value of K3 is given in Table 3 for various Reynolds numbers. The ratio of half-width spread to length scale (K2u/K3) is found to be approximately constant at 1.9 except for low Reynolds number of Re = 250 and 550, where it is 2.5. Such constant ratios are also reported by Cafiero et al. [27] up to streamwise locations of 50d. This shows that the evolution of spread and length scale is similar except at low Re. This is consistent with the previous observations (Figure 2) that only low Re jets are influenced by heating compared to high Re jets (Table 4).
Single and Double Flow Pulsations of Normal and Inverse Partially Premixed Methane-Air Flames
Published in Combustion Science and Technology, 2020
A. M. Hamed, A. M. Moustafa, M. M. Kamal, A. E. Hussin
It is true that the temperature reached across the non-premixed flame is higher than what is reached across the premixed flame by a magnitude of difference which increases as the mixture equivalence ratio is enlarged. It follows that the density experiences a drop across each flame such that there is a corresponding velocity jump (which is more significant across the non-premixed flame when Ф = 2.5). Because a higher velocity is thus reached across the non-premixed flame (in comparison to that of the premixed flame), the velocity fluctuation U’ is higher across the non-premixed flame (Figure 7a). Subsequently, higher amplitudes of fluctuation are noticed in the temperatures downstream of the non-premixed flame reaction zone where the higher velocities cause eddies with higher momentum values. Duplicating the frequency enlarges the strain rates (Figure 7b). In this regard, for the same average velocity (which is 0.637 of the peak velocity) the higher frequency causes a faster velocity change (via slope 2 which is higher than slope 1). Straining the flow by a higher temporal velocity gradient subsequently causes higher spatial velocity gradients in the nearby low velocity regions. These, in turn, increase the flow shearing stresses and set higher turbulent kinetic energies and favorable combustion characteristics. The premixed flame wrinkling increases its surface area for higher QR. The non-premixed flame responds to the higher frequencies by ensuring higher heat release rates. The T.K.E. variation with wave number (Figure 7c) has slopes close to −5/3. The peak energy is reached at the integral scale which is in the order of the local jet width (Lee, Abark, Woo 2015). For U = 3.5 m/s, the de-correlation time scale decreases from 0.040 to 0.032 seconds upon increasing the frequency from 2 to 18 Hz (Figure 7d). As thus more rapidly varying velocity prevails, the turbulence integral length scale decreases from 14.0 to 11.2 cm inside the 15 cm radius combustor. Peak Reynolds stress thus exists at the mixture jet-air interface (Figure 7e).
Turbulent flow characteristics over forward-facing obstacle
Published in Journal of Turbulence, 2021
Debasmita Chatterjee, B. S. Mazumder, Subir Ghosh, K. Debnath
When the incoming flow passes over the obstacle, turbulent length scale with eddies changes along the flow. The turbulent length scales (eddy turn-over scales) are fully dependent on the effective flow depth heff related to the length and height of the submerged obstacle. The integral length scale is a measure of the largest eddy size in the turbulent flow passing over the obstacle. The integral length scale Lu is determined using the relation proposed by [49,50] as follows: where P(f) is the spectral energy as frequency f approaches zero, is the standard deviation of the stream-wise velocity, and is the stream-wise mean velocity. The stream-wise turbulent integral length scales are calculated at the mid-depth along eight locations upstream to downstream, and presented in Table 5. The non-dimensional turbulent length scale is defined as , where is the effective flow depth due to submerged obstacle from the locations A to H with dipped water depth about h′ = 1 cm. From the trend of turbulent length scale Lu, it is ensured that the length scales Lu change upstream to downstream with largest length scale over the obstacle. Moreover, largest length scale 12.99 cm is observed at location D above the crest and it decreases with a similar order of magnitude up to the location F, and then it reduces further downstream (Table 5), and it is grossly related to the height above the bed surface, which is similar to [57]. This phenomenon of transfer of energy may be attributed due to the highest stream-wise velocity above the crest, and then slowly decayed further downstream. Therefore, it is noticed that how the turbulent integral length scales vary in flow regime due to the presence of submerged obstacle in the middle of flume. Moreover, the stream-wise turbulent integral length scales are also computed by auto-correlation method [68,69]. The computed turbulent integral length scales using these two methods are quite similar in magnitudes. Therefore, it is clear how the eddy size changes due to such profile (structure) when a flow is set over it.