California State University, Chico CE 150 Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico CE 150

Elementary Fluid Dynamics Reading: Munson, et al., Chapter 3 CE 150

Inviscid Flow In this chapter we consider “ideal” fluid motion known as inviscid flow; this type of flow occurs when either 1)   0 (only valid for He near 0 K), or 2) viscous shearing stresses are negligible The inviscid flow assumption is often valid in regions removed from solid surfaces; it can be applied to many problems involving flow through pipes and flow over aerodynamic shapes CE 150

Newton’s 2nd Law for a Fluid Particle CE 150

Newton’s 2nd Law for a Fluid Particle The equation of motion for a fluid particle in a steady inviscid flow: We consider force components in two directions: along a streamline (s) and normal to a streamline (n): CE 150

Newton’s 2nd Law Along a Streamline Noting that we have: CE 150

Newton’s 2nd Law Along a Streamline Integrating along the streamline: If the fluid density  remains constant This is the Bernoulli equation CE 150

Newton’s 2nd Law Across a Streamline A similar analysis applied normal to the streamline for a fluid of constant density yields This equation is not as useful as the Bernoulli equation because the radius of curvature of the streamline is seldom known CE 150

Physical Interpretation of the Bernoulli Equation Acceleration of a fluid particle is due to an imbalance of pressure forces and fluid weight Conservation equation involving three energy processes: kinetic energy potential energy pressure work CE 150

Alternate Form of the Bernoulli Equation Pressure head (p/g) - height of fluid column needed to produce a pressure p Velocity head (V2/2g) - vertical distance required for fluid to fall from rest and reach velocity V Elevation head (z) - actual elevation of the fluid w.r.t. a datum CE 150

Bernoulli Equation Restrictions The following restrictions apply to the use of the (simple) Bernoulli equation: 1) fluid flow must be inviscid 2) fluid flow must be steady (i.e., flow properties are not f(t) at a given location) 3) fluid density must be constant 4) equation must be applied along a streamline (unless flow is irrotational) 5) no energy sources or sinks may exist along streamline (e.g., pumps, turbines, compressors, fans, etc.) CE 150

Using the Bernoulli Equation The Bernoulli equation can be applied between any two points, (1) and (2), along a streamline: Free jets - pressure at the surface is atmospheric, or gage pressure is zero; pressure inside jet is also zero if streamlines are straight Confined flows - pressures cannot be prescribed unless velocities and elevations are known CE 150

Mass and Volumetric Flow Rates Mass flow rate: fluid mass conveyed per unit time [kg/s] where Vn = velocity normal to area [m/s]  = fluid density [kg/m3] A = cross-sectional area [m2] if  is uniform over the area A and the average velocity V is used, then Volumetric flow rate [m3/s]: CE 150

Conservation of Mass “Mass can neither be created nor destroyed” For a control volume undergoing steady fluid flow, the rate of mass entering must equal the rate of mass exiting: If  = constant, then CE 150

The Bernoulli Equation in Terms of Pressure Each term of the Bernoulli equation can be written to represent a pressure: pgh : this is known as the hydrostatic pressure; while not a real pressure, it represents the possible pressure in the fluid due to changes in elevation CE 150

The Bernoulli Equation in Terms of Pressure p : this is known as the static pressure and represents the actual thermodynamic pressure of the fluid CE 150

The Bernoulli Equation in Terms of Pressure The static pressure at (1) in Figure 3.4 can be measured from the liquid level in the open tube as pgh : this is known as the dynamic pressure; it is the pressure measured by the fluid level (pgH) in the stagnation tube shown in Figure 3.4 minus the static pressure; thus, it is the pressure due to the fluid velocity CE 150

The Bernoulli Equation in Terms of Pressure The stagnation pressure is the sum of the static and dynamic pressures: the stagnation pressure exists at a stagnation point, where a fluid streamline abruptly terminates at the surface of a stationary body; here, the velocity of the fluid must be zero Total pressure (pT) is the sum of the static, dynamic, and hydrostatic pressures CE 150

Velocity and Flow Measurement Pitot-static tube - utilizes the static and stagnation pressures to measure the velocity of a fluid flow (usually gases): CE 150

Velocity and Flow Measurement Orifice, Nozzle, and Venturi meters - restriction devices that allow measurement of flow rate in pipes: CE 150

Velocity and Flow Measurement Bernoulli equation analysis yields the following equation for orifice, nozzle, and venturi meters: Theoretical flowrate: Actual flowrate: CE 150

Velocity and Flow Measurement Sluice gates and weirs - restriction devices that allow flow rate measurement of open-channel flows: CE 150

Velocity and Flow Measurement CE 150