Fluid Dynamics

Fluid Dynamics:

It deals with the study of fluid(invicid) particles when they are in motion. It involves concepts of force, acceleration and newton's 2nd law of motion. This means, it is the study of fluid flow.

There are two types of flow, namely:

• Steady Flow  ⟼  If the fluid velocity at a point is not a function of time like V = f(x, y, z).
• Unsteady Flow  ⟼  If the fluid velocity changes with time like V = f(x, y, z, t). 

Approaches To Find Fluid Movement:

There are two types of approach which can tells the behavior and movement of fluid particles which are described below:

1. Eulerian Approach:

Eulerian approach depends upon the fixed space that means it depends on flow domain or control volume. If we take velocity field, then Eulerian approach is represented as V = f(x, y, z, t).

2. Lagrangian Approach:

Lagrangian approach depends upon individual particles (for e.g balling styles & its projectile). If we take velocity field, then Lagrangian approach is represented as V = f(x(t), y(t), z(t), t).

Acceleration Field or Material Particle Derivative:

We often say material particle while studying fluid particle. By applying the Lagrangian approach on the material particle, we get:

Note: If you want know how velocity & acceleration field graph are drawn then click on the link given below: Graph and Video.

Flow Patterns & Flow Visualization:

Flow visualization is the visual examination of flow field features and is visualized by a software named Computational Fluid Dynamics (CFD).

There are four basic types of line patterns which are used to visualize flow:

• Streamline  ⟾  It is a line which is obtain by joining tangents to the velocity vectors.

• Path-Line  ⟾  Actual path traversed by fluid particle like google maps and DRS.

• Streak-Line  ⟾  It is the locus of particles pass through any definite point.

• Time-Line  ⟾  Set of lines of fluid particles joining streak-lines with respect to time.

 Euler's Equation Of Motion:

Consider the flow of fluid particles which is moving under the action of gravity having length 'ds' and cross-sectional area 'dA'. 

The Second equation of Motion for differential mass is given by:

Assumptions During Euler Equation of Motion:

There are five assumptions during the derivation of Euler's equation of motion:

1. Invicid (Negligible Viscosity)
2. Steady state
3. Incompressible fluid
4. Along the streamlines 
5. No heat, work or frictional losses

Conversion Of Euler Equation:

We convert Euler equation into energy, pressure and heads equation which are given below:

1. Energy Equation or Bernoulli Equation:

2. Pressure Equation:

3. Head Equation:

Cases of Bernoulli Equation:

A. Simple Free-jet Case (Const. Pressure & Height Vary):

B. Free-jet With Different Diameters:

Torricelli's Theorem Equation:

Torricelli's Theorem states that: The speed of the flowing liquid 'V' under the force of gravity out of an opening in a tank is proportional jointly to the square root of the vertical height multiply the twice the gravitational acceleration.

C. Horizontal Nozzle (Sharp Edge Corners):

If the exit flow is interrupted by sharp corners, the diameter of jet 'dj' will be less than the diameter of hole 'dh'. This phenomena is called Vena Contracta Effect.

Since, the diameter of the jet has changed so the flow rate should be decreased by the value Coefficient of Contraction (CC). 

The Stagnation Point:

When a fluid flowing through a body at rest, the velocity of fluid particles on stagnation point is zero and its pressure is given by:

Bernoulli Equation For Flow Rate Measurement:

To calculate the flow rate 'Q', we use three types of plates or tubes which are as follows:

1. Orifice Tube:

• Uniform diameter pipe with a plate having a hole is placed into it.
• The purpose of hole is to increase pressure difference.
• The velocity of fluid after hole is greater than the fluid velocity before hole.

2. Nozzle Plate:

• Uniform diameter pipe with curve plate placed inside.
• Nozzle (curve like plates) are used to increase pressure differences.

3. Venturi Plate:

• Non-uniform plate having two parts: one is convergence section and the other is divergence section.
• Venturi is used to calculate pressure difference.

Applying Bernoulli Equation, we get:

Energy Lines:

The set of lines showing energies whether it is pressure and kinetic energy are called Energy Lines.

• It is a useful interpretation of Bernoulli equation which can be obtained through the use of energy lines.

There are two types of energy lines which are described as follows:

1. Hydraulic Grade Line (HGL)  ⇒  The line which represent the sum of the static pressure and the elevation heads.
2. Energy Grade Line (EGL)  ⇒  The line which represent the total head of the fluid.

The difference between Hydraulic grade line and Energy grade line is called Dynamic Head.  

Conservation Of Momentum In Fluid Dynamics:

Law of conservation of momentum states that: The total momentum of an isolated system always remain constant.

In fluid dynamics, we use Newton's 2nd Law of Motion:

By separating and differentiating the above equation, one can get:

References:

• Materials From Class Lectures + Own Knowledge + Book named Fundamentals of Fluid Mechanics by Munson, Young and Okiishi's (8th Edition).
• Photos from Google Images.
• Videos from YouTube + Google.