Measurements in Fluid Mechanics 058:180:001 (ME:5180:0001) Time & Location: 2:30P - 3:20P MWF 218 MLH Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui

2 Lecture 2. Description of flows

3 Flow velocity and velocity fields Cartesian coordinate system (x 1, x 2, x 3 ) x1x1 x2x2 x3x3 Position of fluid element n at initial time t 0 : X n,0 X n,0 X n ( t ) Trajectory of fluid element Position of fluid element at time t : X n (t) X n (t)+  X n XnXn Displacement  X n in  t Position after displacement: X n (t+  t)=X n (t)+  X n Flow velocity: Acceleration: Lagrangian description Velocity field: V = V n (t) = V(X n,0,t) for n=1,2,3, ,N

4 Velocity field: U i =U i (x 1, x 2, x 3, t) i=1,2,3 Flow velocity and velocity fields Eulerian ( spatial ) description Acceleration: Local accelerationConvective acceleration Rate of strain tensor: Vorticity:

5 Analytical description of flows Four basic axiomatic principles applied to fixed control volume - The conservation of mass (or continuity equation) - The momentum equation - The first law of thermodynamics - The second law of thermodynamics  - fluid density v - control volume A - area of surface u - specific internal energy p - pressure g - gravitational acceleration s - specific entropy T - temperature z – upwards vertical axis

6 Analytical description of flows - Solid boundary conditions no-penetration condition: (V f –V s ) n = 0 no-slip condition: (V f –V s ) t = 0 V f - fluid velocity V s - solid body velocity ( ) n - normal component ( ) t - tangential component Other important relationships - Two-phase flow interface condition  p - pressure difference cross interface  - surface tension R 1, R 2 - principal radii of curvature - Perfect-gas law R - gas constant - Laws of chemical reaction for reactive flows - Magnetohydrodynamic laws for electrically conductive fluids in magnetic fields - Stress-strain rate relationships for non-Newtonian fluids - Turbulent models - Others

7 Commonly used theoretical approach 1. Fluid statics - assume a fluid is at rest or in rigid-body motion. - No deformation- No shear stresses - Three normal stresses are equal to the pressure - Hydraulic pressure and buoyancy fore caused by gravity - Static fluid analysis used many instruments, e.g. manometers, barometers, and pressure transducers - Static fluid analysis leads to substantial errors when fluids are in motion 2. Inviscid incompressible flows - simplest mathematical model of fluid flow - Neglect effects of friction and compressibility - Continuity equation: - Momentum (Euler) equation: - Bernoulli’s equation: - Assumption of irrotationality – potential flow

8 Commonly used theoretical approach 3. Viscous incompressible flows - with constant density. - Navier Stokes equations - kinematic viscosity - Reynolds number - ratio of inertia forces to viscous forces V – characteristic velocityd – characteristic length  – dynamic viscosity (  =  ) Low-Re flows: stableHigh-Re flows: unstable & turbulent 4. Compressible flows - with significant density variation - Friction often neglected to simplify analysis of compressible flow - Mach number used to describe effects of compressibility V – flow velocityc – speed of sound M<<1: incompressible M>1: supersonic flow M<1: subsonic flow Re  : viscous effects negligible

9 Commonly used theoretical approach 5. Turbulent flows - flow properties vary rapidly and randomly in space and time. - Require statistical description: mean + fluctuation (Reynolds decomposition) - Turbulent intensity: (root-mean-square velocity fluctuation) / (mean velocity) - Measurement of turbulent flow require special instrumentation with refined spatial, temporal and amplitude resolutions - Special methods to resolve turbulent patterns e.g. phase averaging and conditional sampling 6. Complex flows - Multiphase flows - Flows with chemical reaction - Low-density (rarefied) gas flows - Flows of non-Newtonian fluids – nonlinear stress-strain rate relationships - Magnetohydrodynamic flows - Others

10 Homework -Questions and Problems: 4, 7 and 8 on page 17 - Read textbook on page Send MS Word or PDF file to - Due on Friday, 08/26