������������ ������������ = ������������ ������������
⑩
������ = ������ + ������������
������������ ������������
where ������������ is called the eddy viscosity. Fluid viscosities (������) are true properties of the fluid, and can be measured on isolated samples of the fluid; eddy viscosity (������������ ) depends on both the fluid and also the fluid velocity and system geometry and is sensitive to location in the turbulent field and the
������������������ inertial force ������������ = = ������ Viscous force
Parabolic velocity distribution, with
������ = ������������,������������������ ������ Shear stress is linear in radial distance Volumetric flow rate is given by the Hagen-Poiseuille equation The Fanning friction factor is given by ������ = ������������ ������������ ℎ������������
������������������ 4 ������������������4 ������ = = 128������������ 8������������
© 2015 Yanwei Wang
Turbulent flow in pipes
The onset of turbulence: The top panel shows an image of a single “puff” (localized turbulent patches embedded in the surrounding laminar flow) structure, at a Reynolds number of 2000 (just below the onset of turbulence). As the Reynolds number increases, the spatial scale of the structure decreases.
6 © 2015 Yanwei Wang
Interim summary on turbulent flow (1/4)
① Occurs at high Re (Re > 4000 for flow in a pipe) ② In Reynolds experiment, dye mixes rapidly and completely – Hence, turbulent flow is important for mixing. ③ Path of fluid particles completely irregular, tends to produce chaotic eddies, vortices and other flow instabilities. ④ Deviating velocities: Average motion is in the direction of
the flow, but there are rapid fluctuations in local velocities,
often characterized by statistical analysis. ⑤ Most common type of flow, but Mathematical analysis very
Layer flow with free surface
Incompressible flow in pipes and channels Lamw)
2 © 2015 Yanwei Wang
Homework (HW-6)
HW6-1: Reynolds number and transition from laminar to turbulent flow: (a) airflow and (b) water flow. HW6-2: Incompressible flow through an inclined pipe
5
Δ������������2 ������ ������������ ������ = 1− 4������������ ������ ������������������ ������ = Δ������ 2������
2
∆������������ 4 ������ ������ ������ 2 ������ ������ 2 = = ������������ = 4������ = ������ ������ ������ ������ ������ 2 ������ 2
2100 and 4000 a transition region
is found where the flow may be either laminar or turbulent, The Onset of Turbulence in Pipe Flow. ������ is the mean lifetime of a puff before decaying or splitting. Avila, K. at al., Science, 333, pp.192 (2011)
local values of the scale and intensity of turbulence. Difficult to
determine in practice.
9
© 2015 Yanwei Wang
Interim summary on Turbulence (4/4)
The transition from laminar to turbulent flow may occur over a wide range of Re values. In a pipe, flow is always laminar at Re < 2100, Under ordinary conditions, the flow in a pipe or tube is turbulent at Re > 4000. Between
Sketch for HW6-2
3 © 2015 Yanwei Wang
Today’s topic
Date: March 23, 2015
Pages 29-33 and 52-56 of textbook
Turbulent flow in pipes and channels
Velocity profile
Pages 25-29 & 43-56 of textbook
Types of fluid flow and Reynolds number
Reynolds experiment Laminar vs. Turbulent flow Reynolds number Macroscopic momentum balances
4 © 2015 Yanwei Wang
Summary of Laminar flow in pipes
Poiseuille flow, Re < 2100
Total shear stress is the viscous stress given by Newton’s law of viscosity
Relation between ������������������������ and ������
Effect of roughness Moody’s chart
Boundary layers
Understand the concept
Be able to explain Figs. 1.8, 1.9, 1.10, and 1.11
Fall Semester, 2014
Basic Principles of Chemical
Engineering Processes
Lecture 7
Yanwei Wang (王衍伟) Email: ywwang@
© 2015 Yanwei Wang
Previous Lecture
8
© 2015 Yanwei Wang
Interim summary on Turbulence (3/4)
⑨ The total shear stress in a turbulent fluid is the sum of the viscous stress and the turbulent stress, ������������
difficult – hence experimental measures are used.
7
© 2015 Yanwei Wang
Interim summary on Turbulence (2/4)
⑥ Turbulent flow consists of eddies of various sizes, but even the smallest eddies have diameter 10 to 100 μm, so turbulent flow is not a molecular phenomenon. ⑦ The turbulent energy cascade: The energy by external forces excites the largest possible eddies and is gradually passed to ever smaller eddies, all the way to a minimum scale where this energy is ultimately dissipated to heat by viscous dissipation.