have dominated engineering practice since the beginning of the 20 th Century

Have dominated engineering practice since the

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have dominated engineering practice since the beginning of the 20 th Century because turbulence is deterministic , not random , and it is described by the N.–S. equations. Moreover, laboratory experiments commencing in the mid 1970s and continuing to the present have repeatedly supported the theory proposed in 1971; namely, the transition to turbulence takes place through a very short sequence (usually three to four) of distinct steps (recall Fig. 2.22). In this sense, the “turbulence problem” has been solved: the solution is the Navier–Stokes equations. But from a practical engineering standpoint this is not of much help. Despite the tremendous advances in computing power and numerical solution techniques that have arisen during the last quarter of the 20 th Century, we still are far from being able to simulate high-Reynolds number turbulent flows that arise on a routine basis in actual engineering problems. Furthermore, this
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4.5. PIPE FLOW 135 is expected to be the case for at least the next 20 years unless true breakthroughs in computing hardware performance occur. This implies that in the near to intermediate future we will be forced to rely on considerable empiricism for our treatments of turbulent flow. It is the goal of the present subsection to provide an elementary introduction to this, specifically as it pertains to flow in pipes of circular cross section, for which voluminous amounts of experimental data exist. It is worthwhile to first present a “way of viewing” turbulent pipe flow (and wall-bounded turbulent flows, in general) in terms of distinct physical regions within such flows. Figure 4.14 provides a simple schematic containing the essential features. To better understand the physics that layer buffer sublayer inertial viscous sublayer pipe centerline Figure 4.14: Turbulent flow near a solid boundary. occurs in each of the regions shown in this figure it is useful to consider a particular representation of the velocity field known as the Reynolds decomposition . This is given in 2D as u ( x, y, t ) = u ( x, y ) + u ( x, y, t ) , (4.49) for the x -component of velocity, with similar formulas for all other dependent variables. In Eq. (4.49) the bar (“ ”) denotes a time average, and prime (“ ”) indicates fluctuation about the time-averaged, mean quantity. Formally, the time average is defined as, e.g. , for the x -component of velocity, u ( x, y ) lim T →∞ 1 T integraldisplay T 0 u ( x, y, t ) dt . (4.50) At any particular spatial location, say ( x, y ), in a stationary flow (one having steady mean proper- ties) the time-averaged and fluctuating quantities might appear as in Fig. 4.15. Now the purpose of introducing the decomposition of Eq. (4.49) is to aid in explaining some additional physics that occurs in turbulent flow, and which is not present in laminar flows. In par- ticular, we consider the advective terms of the N.–S. equations, expressed for the 2-D x -momentum equation as ( u 2 ) x + ( uv ) y , which follows from the divergence-free condition. For the second of these terms we can write ( uv ) y = ( ( u + u )( v + v ) ) y = ( u v ) y + ( uv ) y + ( u v ) y + ( u v ) y , (4.51) with similar expressions available for all other inertial advection terms.
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