Advanced Fluid Mechanics Principles of Fluid Mechanics ➢Ventilation is the application of the principles of fluid dynamics to the flow of air ➢Airflow is induced from the atmosphere, through the intake opening ➢and back to the atmosphere again by a differential pressure between intake and return openings ➢Airflow is also generally treated as steady state, turbulent, and incompressible. ➢However, in situations where the pressure, temperatures, and humidity changes are large and air conditioning processes are involved, calculations must incorporate compressibility effect of air Definition of A Fluid ➢ A fluid may be defined broadly as a substance which deforms continuously when subjected to shear stress . This fluid can be made to flow if it is acted upon by a source of energy ➢ If the neighboring layers offer no resistance to the movement of fluid, this fluid is said to be frictionless fluid or ideal fluid. (Practically speaking, ideal fluids do not exist in nature, but in many practical problems the resistance is either small or is not important, therefore can be ignored.) ❖The properties of a fluid, e.g., density, may, however, vary from place to place in the fluid ❖In addition to shear force, fluid may also be subjected to compressive forces. These compressive forces tend to change the volume of the fluid and in turn its density. ❖If the fluid yields to the effect of the compressive forces and changes its volume, it is compressible, otherwise it is incompressible Turbulent and Laminar Flows ➢ In a flowing fluid, each particle changes its position with a certain velocity. The magnitudes and directions of the velocities of all particles may vary with position as well as with time ➢ Reynold’s number: is an important dimensionless quantity in predict flow patterns in different fluid situations. ➢ Laminar flow: Re<2300 ➢ Transient flow: 2300<Re<4000 ➢ Turbulent flow: 4000<Re Streamlines ➢ A streamline is an imaginary line in a fluid, the tangent to which gives the direction of the flow velocity at that position, as shown in Figure 4-2, where the distance between two streamlines is an inverse measure of the magnitude of the velocity. Streamlines ➢ If the streamlines are smoothly curved and almost parallel to each other, as illustrated in Figure 4-3, the flow is known as streamlined flow or laminar flow ➢ A streamline is an imaginary line in a fluid, the tangent to which gives the direction of the flow velocity at that position, as shown in Figure 4-2, where the distance between two streamlines is an inverse measure of the magnitude of the velocity. ➢ If the streamlines are smoothly curved and almost parallel to each other, as illustrated in Figure 4-3, the flow is known as streamlined flow or laminar flow . ➢ On the other hand, if the streamlines are arranged haphazardly as illustrated in Figure 4-4, the flow is known as turbulent flow . Reynold’s number (NRe) is a dimensionless number Example Equation of motion ➢ According to Newton's Second Law, the force F used to accelerate a body must be equal to the product of the mass m of the body and its acceleration a. Mathematically: Fx = m.ax i. Fg: gravity force ii. Fp: the pressure force iii. Fv: force due to viscosity iv. Ft: force due to turbulence v. Fc: force due to compressibility Fx = (Fg)x + (Fp)x + (Fv)x + (Ft)x + (Fc)x ❖ If the force due to compressibility , “Fc” is negligible the resulting net force Fx = (Fg)x + (Fp)x + (Fv)x + (Ft)x ➢ And the equations of motions are called Reynold’s equations of motion ❖ For flow where (Ft) is negligible the resulting equations of motion are known as Navier-Stokes Equation. ❖ If the flow is assumed to be ideal, viscous force (Fv) is zero and equations of motions are known as Euler’s equation of motion. Euler’s equation of motion Consider a cylindrical element of cross-section dA and length dS. The forces acting on the cylindrical element are: Bernoulli’s equation from Euler’s equation We have got Bernoulli's equation where Example 01 ❑ Water is flowing through a pipe of 5cm diameter under a pressure of 29.43 N/cm2 (gauge) and with a mean velocity of 2.0 m/s. Find the total head or total energy per unit weight of the water at a cross-section, which is 5m above the datum line. Example 02 ❑ A pipe through which water is flowing is having diameters 20cm and 10cm at the cross-section 1 and 2 respectively. The velocity of water at section at section 1 is given 4.0m/s. find the velocity head at sections 1 and 2 and also rate of discharge. Exercise Water is flowing through a pipe having diameter 300mm and 200mm at the bottom and upper end respectively. The intensity of pressure at the bottom end is 24.525 N/cm2 and the pressure at the upper end is 9.81N/cm2. Determine the difference in datum head if the rate of flow through pipe is 40 lit/s Bernoulli’s equation for real fluid ➢ Bernoulli’s equation was derived on the assumption that fluid is inviscid (nonviscous) and therefore frictionless. ➢ But all the real fluids are viscous and hence offer resistance to flow. ➢ Thus there are always some losses in fluid flows and hence in the application of Bernoulli’s equation. ➢ Thus the Bernoulli’s equation for real fluids between points 1 and 2 is given as Example Practical applications of Bernoulli’s equation Venturimeter A venturimeter is a device used to measure the rate of flow of a fluid flowing through a pipe. It consists of three parts: Example Example Moment of momentum equation