2.2 The Turbulent Round Jet
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1 Canonical Turbulent Flows 13. The Turbulent Round Jet Jet flows are a subset of the general class of flows known as free shear flows where free indicates that the shear arises in the absence of a boundary (wall) and instead results from a maintained velocity difference. The scale of round jet flows ranges from the nozzle on an ink jet printer to a volcano. For a perfectly round jet the flow is essentially completely characterized by the Reynolds number Re D = U JD ν (.11) where D is the diameter of the jet orifice and U J is the velocity of the jet at the orifice. Ideally the velocity at the orifice will be constant across the exit plane, but if not we take U J = Q/A where Q is the flow rate and A is the jet cross-sectional area. At low Re D jets are laminar and at high Re D jets are turbulent. It is a bit difficult to define exactly where the transition occurs as it depends on the exact conditions of the flow at the orifice. Jets begin to exhibit turbulent behavior at relatively low Re D, as low as 300, but the fundamental parameters that characterize the turbulent round jet (to be defined in a moment) are weakly a function of Re D and the jet exit velocity profile (Zarruk & Cowen, 008. Simultaneous velocity and passive scalar concentration measurements in low Reynolds number neutrally buoyant turbulent round jets. Experiments in Fluids 44, ). For an essentially uniform jet exit velocity profile (known as a top-hat profile), the fundamental parameters of the turbulent jet become independent of Re for Re D > 4000, thus the canonical turbulent round jet problem is for Re D > Round jets are one of many flows that exhibit what is known as self-similar behavior. Self-similarity indicates that a flow will look similar when scaled appropriately. This appropriate scaling involves defining similarity variables that reduce the overall dimensionality of the problem. In the case of the turbulent round jet, after a transition region that occurs roughly from 0 <x/d<30 (where x is the coordinate along the jet axis coordinate system shown on next page), the jet cross-sectional properties are independent of x when properly scaled. The similarity variable in the round jet can be chosen in multiple ways but basically comes down to a statement that the radial coordinate c 017 Edwin A. Cowen
2 14 Canonical Turbulent Flows depends on the axial coordinate in a fixed way. We will choose as our similarity variable the ratio r/r 1/ where r 1/ is the half-width of the jet defined as the radial point where the velocity has dropped to half the centerline velocity. Several other normalizations have been developed including r/(x x 0 )wherex 0 is a virtual origin to be defined in a moment and r/l q where l q = A. Our choice of definition has the advantage that a single cross-sectional profile can be normalized without exact knowledge of the streamwise position of the profile relative to the jet orifice. For the round jet we define U 0 (x) =u(x, 0, 0) (.1) u(x, r 1, 0) = U 0(x) (.13) where U 0 is our nomenclature for the jet centerline velocity. If the centerline velocity is plotted versus the non-dimensional distance downstream an inverse relationship with downstream distance is found (this can actually be analytically predicted) U 0 (x) B U = ( ) J or U J x x0 U 0 (x) = 1 ( ) x x0 B D D (.14) where B (the slope term of the centerline velocity decay which Hussein, Capp &George (1994) define as B u, see below) and x 0 (the virtual origin of the jet) are constants of the expected linear fit that must be determine empirically for Re D > 4000 B 5.8 while x 0 /D is generally 4 but each are a function of Reynolds number and jet exit profile (Zarruk & Cowen, 008). c 017 Edwin A. Cowen
3 Canonical Turbulent Flows 15 I have handed out Hussein, Capp & George (1994). Velocity measurements in a high- Reynolds-number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 58, Look over this article for plots of Eq.14 and all of the plots below for jets. This is the current standard article on turbulent round jets. The standard paper prior to this paper (and still a benchmark!) was (available as a pdf in the \handouts section of the website) Wyzgnanski & Fiedler (1969). Some measurements in the self-preserving jet. J. Fluid Mech. 38, The same plots exist in this paper. Note both papers are excellent examples of experimental design and technology and should be regarded as exemplary experimental papers. Each reviews hot-wire anemometry and Hussein et al. reviews laser Doppler velocimetry (LDV) as well. We can define the jets spreading rate, S, basedonr 1 as (x) S = dr1 dx (.15) The conservation of momentum can be used to show that in a round jet the product r 1 (x)u 0 (x) is a constant. Eq.14 indicates that U 0 (x) x 1 hence r 1 (x) mustgoasx 1. Integration of Eq.15 and incorporation of the virtual origin results in r 1 (x) =S(x x 0 ) (.16) A typical value for S is (Hussein et al.). Note based on the definition arctan S gives the angle of spread of the r 1 (x) contour. ForS =0.094 the angle with respect to the x axis is about 5.4. A more physical sense of the jet width is based on S (greater than 90% of the momentum is contained within this contour). The included angle of the jet based on S is about 1 (e.g., the angle between the ±S contours). The above analysis is all based on mean quantities how does the turbulence in a round jet behave? It turns out that it too is self-similar. If we consider u and normalize it by the jet centerline velocity, U 0 (x), we find that it also decays as x 1 and in the self-similar region has a constant value of about 7% (Hussein et al.). If we consider v we find the same dependence however the constant value is a bit lower, about % (Hussein et al.). Note that the round jet is axially symmetric (there is no dependence on the exact c 017 Edwin A. Cowen
4 16 Canonical Turbulent Flows direction we choose perpendicular to the jet axis). Hence if we continue our analysis of the turbulence in Cartesian coordinates we find that w has the same dependence and the constant is %. Some general points about the shape of the turbulence profiles in the radial direction are worth noting. If you consider the shape of u it decays monotonically (but in a Gaussian like manner) from the peak centerline value to zero at an infinite radius. There is an inflection point in the profile where the maximum radial gradient occurs (at around r/r 1 = 0.5). This is the point of maximum shear and it is the point where turbulence production is a maximum. Since the shear is in the u component of the velocity field it should not be surprising that u shows a peak value above that of the centerline value at this point in the profile and then decays. As there is minimal shear driven production in the plane perpendicular to the jet axis the v and w profiles show no such offaxis peak but instead decay in a way very similar to u. See Hussein et al. s Figure 9-11 for examples. The final turbulence parameter we need to discuss is the Reynolds stress component u w (which is the same as the u v in the case of the round jet). Recall that this is the turbulent transport of momentum term. Physically the jet has its momentum focused along the jet axis and we expect that the turbulence is trying to spread the momentum radially away from the axis. Thus if we consider the upper half of the jet profile where u/ z <0 a positive w fluctuation will carry high momentum fluid away from the jet axis and hence u > 0. Thus in this region we expect u w > 0. An easy way to get a sense for the Reynolds stress behavior is to use what is called a scatter plot a plot of u vs w in the case just described. Here is an example of a scatter plot from ADV data near the region of maximum shear in a turbulent round jet (e.g., lab #1!). Similarly if we now consider the lower half of the jet profile where u/ z >0 a negative w fluctuation will carry high momentum fluid away from the jet axis and hence u > 0. Thus in this region we expect u w < 0. There will not be a discontinuity at the jet centerline so the value at jet centerline must be u w = 0. The Reynolds stress terms c 017 Edwin A. Cowen
5 Canonical Turbulent Flows w (cm/s u (cm/s play a pivotal role in turbulence production and hence we expect their value to be a peak where the mean velocity shear is a maximum - at around r/r 1 = 0.5 as indicated above. Further, as u(x) /U 0 (x) has a self-similar region we expect that u w /U0 will also be self-similar and it is. See Hussein et al. s Figure 1 for the profiles exact shape. That s it for now for the turbulent round jet. For more details see the two articles handed out and the excellent chapter in Pope (001), sections , on the reference list. We will investigate the mass flux in a turbulent round jet in the final laboratory exercise so the story is not quite over! c 017 Edwin A. Cowen
6 18 Canonical Turbulent Flows c 017 Edwin A. Cowen
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