Ch5_6 - Chapter 5 Free thin shear ows In this Chapter, we...

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Chapter 5 Free thin shear flows In this Chapter, we will discuss a large class of flows called thin shear flows . We will focus on flows far from a solid surface (the wall boundary layer is discussed in Chapter 6) and we will aim to understand the structure of jets , wakes ,and mixing layers .T h i s i sd o n e using: (i) the governing equations; (ii) the eddy viscosity model; (iii) the boundary layer approximation; (iv) the idea of self-similarity, (v) global conservation principles, and (vi) input from experiment. We do not aim for mathematical rigour in the theory, but to bring across the main physical Fndings. Davidson’s Ch. 4 [2] and Tennekes and Lumley’s Ch. 4 [1] have perhaps the most accessible expositions. Pope’s Chapter 5 [3] is also very good and thorough. Here we will present detailed analysis only for the planar and the round jet and we will just quote some results for the other flows. Consult the textbooks for more details. 5.1 The thin shear flow assumption If the mean velocity is predominantly in one direction, then the width of the flow δ is small compared to its length L from its origin, or δ ( x ) /x ± 1; ±ig. 5.1. This has two very important implications. ±irst, that streamwise gradients are much smaller than the cross- stream gradients, and second, that the cross-stream momentum equation can be simpliFed. Both of these result in a very simpliFed form of the streamwise momentum equation, which is amenable to analysis. In this Chapter, we will take as u and v the streamwise and cross-stream velocities, so that u = u x , v = u y , u 0 = u 0 x v 0 = u 0 y . L turb will be a characteristic large-eddy lengthscale, of the order of the flow width δ . We will assume constant pressure in the x direction throughout and high Reynolds numbers, so the visous stresses will drop out of the momentum equations. The various flows that fall under this category of thin shear flows are described in more detail next. 5.1.1 Jets Qualitative description The flow exits a nozzle at velocity U 0 into stagnant fluid, ±ig. 5.1. The nozzle has size d , which is the diameter in the round (or axisymmetric) jet and the slit width in the planar jet. In the latter, the flow is homogeneous in the z direction. The 35
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x r,y U 0 U c ( x ) G 1/2 ( x ) p f ( x ) JET (plane or round) WAKE (plane or round) x r,y U 0 U c ( x ) 1/2 ( x ) Figure 5.1: The flow patterns of jets and wakes. 36
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SHEAR OR MIXING LAYER G 1/2 ( x ) U 1 U 2 x y U c =U 1 0 U 2 Figure 5.2: The flow pattern in a mixing layer. jet then expands and the jet fluid mixes with the ambient fluid. Turbulence is produced by the shear induced by the velocity diference between the jet and the ambient and it is this turbulence that is causing the jet expansion. The thin shear flow assumption implies that the angle o± the jet is small. There is an inter±ace between the turbulent flow in the jet and the non-turbulent flow in the ambient, which moves slowly outwards as the jet engul±s more ambient fluid. The total momentum flow rate in the jet is constant, but the mass flow rate is not, since the jet entrains mass ±rom the surroundings.
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This note was uploaded on 09/16/2011 for the course ME 563 taught by Professor Staff during the Spring '11 term at Auburn University.

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Ch5_6 - Chapter 5 Free thin shear ows In this Chapter, we...

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