Page 188 - Bird R.B. Transport phenomena
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172  Chapter 5  Velocity Distributions in Turbulent Flow











                                                            Fig. 5.6-3.  Streamline pattern  in a circular jet in
                                                            turbulent  flow  [H. Schlichting, Boundary-Layer The-
                                                            ory, McGraw-Hill, New York, 7th edition (1979),
                                                            Fig. 24.10].




                           eddy  viscosity  method  seems  to be somewhat  better  in  the neighborhood  of  the  maximum,
                           whereas the mixing length results are better in the outer part  of the jet.
                               Once the velocity profiles  are known, the streamlines can be obtained.  From the stream-
                           lines, shown in Fig. 5.6-3, it can be seen how the jet draws in fluid  from the surrounding mass
                           of fluid. Hence the mass  of fluid  carried by the jet increases with the distance from the source.
                           This mass rate of flow is

                                                    w =      pv zr dr de = 8TTP*/°Z             (5.6-25)

                           This result corresponds to an entry in Table 5.1-1.
                               The two-dimensional jet issuing from a thin slot may be analyzed similarity. In that prob-
                           lem, however, the turbulent viscosity is a function  of position.



                           QUESTIONS     FOR DISCUSSION

                        1.  Compare and  contrast the procedures  for  solving laminar  flow  problems and  turbulent  flow
                           problems.
                        2.  Why must Eq. 5.1-4 not be used  for evaluating the velocity gradient at the solid boundary?
                        3.  What does the logarithmic profile  of Eq. 5.3-4 give for the fluid  velocity at the wall? Why does
                           this not create a problem in Example 5.5-1 when the logarithmic profile  is integrated  over the
                           cross section of the tube?
                        4.  Discuss the physical interpretation  of each term in Eq. 5.2-12.
                        5.  Why is the absolute value sign used in Eq. 5.4-4? How is it eliminated  in Eq. 5.5-5?
                        6.  In Example 5.6-1, how do we know that the momentum  flow through any plane  of constant z
                           is a constant? Can you imagine a modification  of the jet problem  in which that would  not be
                           the case?
                        7.  Go through some of the volumes oi Ann. Revs. Fluid Mech. and summarize the topics in turbu-
                           lent flow that are found  there.
                        8.  In Eq. 5.3-1 why do we investigate the functional  dependence  of the velocity gradient  rather
                           than the velocity  itself?
                        9.  Why is turbulence such a difficult  topic?


      PROBLEMS        5A.1  Pressure  drop needed  for  laminar-turbulent  transition.  A fluid  with  viscosity  18.3 cp and
                           density  1.32 g/cm 3  is flowing  in a long horizontal  tube  of radius  1.05 in.  (2.67 cm). For what
                           pressure gradient will the flow become turbulent?
                                                  5
                           Answer: 42 psi/mi  (1.8 X 10  Pa/km)
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