Answer all parts of the following: a) The mass and linear momentum balance equations for Newtonian...

  

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Answer all parts of the following: a) The mass and linear momentum balance equations for Newtonian incompressible fluids of density p and (constant) viscosity read: Iv V. v=0 == -pv (vv) - Vp + V Vv+pg t where and p are the fluid velocity and pressure, respectively, while g is the gravitational field. Solving these equations (along with suitable initial and boundary conditions) yields the velocity and pressure profiles. A student carries out an experiment involving liquid water flowing in a pipe. The flow is turbulent. She also simulates the (same) experiment using a computational fluid dynamics code, solving the equations numerically and obtaining the velocity and pressure profiles in the pipe. To test the validity of the equations of motion, she measures experimentally the fluid velocity in various points in the pipe at different times and compares the values with those obtained numerically. She finds that the values do not match. She is quite surprised, because the error is considerable. Is the outcome of this test really surprising? If you think it is not, explain why. d) Outline the bases of the Prandtl mixing length theory of turbulence. Show that for a turbulent shear flow the order of magnitude of the turbulent viscosity t can be estimated as follows: d Ht ~ PL 12/1/1 (0%) / e) where p is the fluid density, L is the length scale of the mean flow, (vx) is the x-component of the mean fluid velocity, x is the direction in which the fluid flows and y is the direction in which (vx) varies. [10] Water flows with mean velocity V = 1 m s in a pipe of diameter D = 0.1 m. Its kinematic viscosity is v = 10-6 m s-1, while the velocity scale of the smallest eddies in the flow is v = 0.1 m s-1. Determine the characteristic time of the smallest eddies and show that it is far less than the time that characterizes the mean flow. What does this large separation of time scales indicate? [6] Answer all parts of the following: a) The mass and linear momentum balance equations for Newtonian incompressible fluids of density p and (constant) viscosity read: Iv V. v=0 == -pv (vv) - Vp + V Vv+pg t where and p are the fluid velocity and pressure, respectively, while g is the gravitational field. Solving these equations (along with suitable initial and boundary conditions) yields the velocity and pressure profiles. A student carries out an experiment involving liquid water flowing in a pipe. The flow is turbulent. She also simulates the (same) experiment using a computational fluid dynamics code, solving the equations numerically and obtaining the velocity and pressure profiles in the pipe. To test the validity of the equations of motion, she measures experimentally the fluid velocity in various points in the pipe at different times and compares the values with those obtained numerically. She finds that the values do not match. She is quite surprised, because the error is considerable. Is the outcome of this test really surprising? If you think it is not, explain why. d) Outline the bases of the Prandtl mixing length theory of turbulence. Show that for a turbulent shear flow the order of magnitude of the turbulent viscosity t can be estimated as follows: d Ht ~ PL 12/1/1 (0%) / e) where p is the fluid density, L is the length scale of the mean flow, (vx) is the x-component of the mean fluid velocity, x is the direction in which the fluid flows and y is the direction in which (vx) varies. [10] Water flows with mean velocity V = 1 m s in a pipe of diameter D = 0.1 m. Its kinematic viscosity is v = 10-6 m s-1, while the velocity scale of the smallest eddies in the flow is v = 0.1 m s-1. Determine the characteristic time of the smallest eddies and show that it is far less than the time that characterizes the mean flow. What does this large separation of time scales indicate? [6]