Fluid Dynamics Notes
Fluid Dynamics
Fluid dynamics is the study of fluids in motion relative to other parts.
Types of Flow
Several types of flow exist:
Steady and Unsteady flow
Uniform and non-uniform flow
Compressible and incompressible flow
Rotational and Irrotational flow
1-D, 2-D and 3-D flow
Potential flow
Laminar and Turbulent flow
Steady and Unsteady Flow
Steady flow occurs when flow conditions (velocity, pressure, density) at a point do not change with time.
Unsteady flow occurs when flow characteristics at a point change with time.
Uniform and Non-Uniform Flow
Uniform flow: No variation in velocity's magnitude and direction from one point to another along the flow path; velocity does not change along the length of the flow.
Non-uniform flow: Velocity of flow doesn't remain constant at all points in space during a given interval.
Rotational and Irrotational Flow
Rotational flow: Fluid particles rotate about their own mass axis while flowing along streamlines.
Irrotational flow: Fluid particles do not rotate about their own axis while flowing along streamlines.
Compressible and Incompressible Flow
Compressible flow: Density of the fluid changes from point to point.
Incompressible flow: Density of the flowing fluid is constant.
1-D, 2-D, and 3-D Flows
One dimensional (1-D) flow: Flow parameter (velocity) is a function of time and one space co-ordinate. u = f(x, t)
Two dimensional (2-D) flow: Flow parameter (velocity) is a function of time and two rectangular space co-ordinates. u = f(x, y, t)
Three dimensional (3-D) flow: Flow parameter (velocity) is a function of time and three mutually perpendicular space coordinates. u = f(x, y, z, t)
Potential Flow
Moving fluid uninfluenced by stationary solid walls is not subjected to shear, and shear stresses do not exist within it. The flow of incompressible fluid with no shear is called potential flow.
Characteristics:
No circulation or eddies can form within the stream; also called irrotational flow.
No friction develops, so there is no dissipation of mechanical energy into heat.
Laminar, Turbulent, and Transitional Flows
Laminar flow: Fluid particles move along well-defined paths or stream lines; streamlines are straight and parallel. Also called stream line flow or viscous flow. Occurs at low velocities and/or high viscosities.
Turbulent flow: Particles do not restrict themselves to their own laminar but continuously change their layers.
Transitional flow: Change over from laminar flow to turbulent flow or vice versa.
Streamline and Stream tube
Streamline: Path traced by a mass-less particle moving along the flow. Velocity at any point is represented by an arrow showing the relative velocity and direction of the flow. Streamlines do not intersect.
Stream tube: An imaginary pipe in the mass of flowing fluid through the walls of which no net flow is occurring.
Reynolds Number
Re = {rho}VL/{\mu} = VL/{\nu} = Inertia force/Viscous force
Where:
V = Velocity of fluid, m/s
L = Characteristic length, m (length of plate (L) or diameter of pipe (D))
{rho} = density
{mu} = viscosity
Boundary Layer
Boundary layer is the region near a solid surface where the fluid motion is affected by the solid boundary.
Entry Length
Entrance region where a nearly inviscid upstream flow converges and enters the tube. Viscous boundary layers grow downstream, retarding the axial flow v(x, r) at the wall and thereby accelerating the center-core flow to maintain the incompressible continuity requirement.
Length up to which flow becomes fully developed.
Boundary Layer Separation and Wake Formation
Boundary layer separation occurs whenever the change in velocity of the fluid, in either magnitude or direction, is too large for the fluid to adhere to the solid surface.
Sharp corners almost always produce a separated flow.
Most frequently encountered when there is an abrupt change in the flow channel such as sudden expansion or contraction, a sharp bend, or an obstruction around which the fluid flows.
Wake Formation
Sharp corners typically produce a separated flow, but they are not the only cause of separation.
Flow over a cylinder can produce a large region of separated flow downstream of the cylinder. This region is called a wake.
Flow patterns over the front and back of the cylinder are quite different. In the front, the flow smoothly passes over the cylinder, but in the wake the flow is usually highly unsteady, and large eddies or vortices are shed downstream.
Basic Equations of Fluid Flow
Fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion. It has several sub disciplines itself, including:
Aerodynamics (the study of air and other gases in motion) and
Hydrodynamics (the study of liquids in motion).
Fluid dynamics has a wide range of applications:
Calculating forces and moments on aircraft,
Determining the mass flow rate of petroleum through pipelines
Predicting weather patterns
Continuity Equation
A1V1 = A2V2
Bernoulli's Equation
In ideal incompressible fluid, the sum of potential energy, kinetic energy, and pressure energy is constant.
Limitations of Bernoulli's Equation:
Steady flow
Frictionless flow
No shaft work
Incompressible flow
No heat transfer
Applied along stream line
Correction in Bernoulli’s equation
Head loss (h_L) is considered:
\frac{P1}{{rho}g} + \frac{V1^2}{2g} + Z1 = \frac{P1}{{rho}g} + \frac{V2^2}{2g} + Z2 + h_L
h_L = Head loss in height of fluid column.
If pump work (W_{pump}) is considered,
\frac{P1}{{rho}g} + \frac{V1^2}{2g} + Z1 + {\eta}{pump}W{pump} = \frac{P1}{{rho}g} + \frac{V2^2}{2g} + Z2 + h_L
Navier-Stokes Equation
Navier-Stokes equation, in fluid mechanics, a partial differential equation that describes the flow of incompressible fluids.
Navier-Stokes Equations
Continuity Equation
\nabla \cdot V = 0
Momentum Equations
{rho} \frac{DV}{Dt} = -\nabla p + {rho} g + {\mu} \nabla^2 V
Total derivative
\frac{DV}{Dt} = [\frac{{partial}V}{{partial}t} + (V \cdot \nabla)V]
Change of velocity with time
Convective term
Pressure gradient
Fluid flows in the direction of largest change in pressure.
Body force term
External forces that act on the fluid (gravitational force or electromagnetic).
Diffusion term
For a Newtonian fluid, viscosity operates as a diffusion of momentum.
Hydraulic Gradient Line (HGL)
Hydraulic gradient line is basically defined as the line which will give the sum of pressure head and datum head or potential head of a fluid flowing through a pipe with respect to some reference line.
Hydraulic gradient line = Pressure head + Potential head or datum head
HGL = \frac{P}{{rho}g} + Z
\frac{P}{{rho}g} = Pressure head
Z = Potential head or datum head
Total Energy Line (TEL)
Total energy line is basically defined as the line which will give the sum of pressure head, potential head, and kinetic head of a fluid flowing through a pipe with respect to some reference line
Total Energy Line = Pressure head + Potential head + Kinetic head
TEL = \frac{P}{{rho}g} + Z + \frac{V^2}{2g}
\frac{V^2}{2g} = Kinetic head or velocity head
Relation between hydraulic gradient line and total energy line
HGL = TEL - \frac{V^2}{2g}
Pipes in Series
Pipes are said to be in series if they are connected end to end (in continuation with each other) so that the fluid flows in a continuous line without any branching.
The volume rate of flow through the pipes in series is the same throughout.
Pipes in Parallel
Pipes are said to be in parallel when they are so connected that the flow from a pipe branches or divides into two or more separate pipes and then reunite into a single pipe.
In this arrangement the total discharge Q divides into components Q1 and Q2 along the branch pipes such that,
Q = Q1 + Q2
In this arrangement the loss of head from section 1-1 to section 2-2 is equal to the loss of head in any one of the branch pipes.
hf = h{f1} = h_{f2}
Flow over Aerofoil
Aerofoil is also called an airfoil. It is a surface shaped like an airplane wing, tail, or propeller blade, that produces lift and drag when moved through the air.
An aerofoil generates a lifting force that acts at right angles to the airstream and a dragging force that acts in the same direction as the airstream.
Lift and Drag
Lift is defined as the component of the aerodynamic force that is perpendicular to the flow direction. Lift conventionally acts in an upward direction in order to counter the force of gravity.
Drag is the component of the aerodynamic force parallel to the flow direction. Drag force acts opposite to the relative motion.
Venturimeter
Applying Bernoulli Energy equation analyzing, the volume flow rate can be obtained from following equation,
A1V1 = A2V2
Multiply by Area at throat we can find flow rate, Q_{thr}
Actual flow rate, Qa = Cd x Q_{thr}
\frac{P1}{{rho}g} + \frac{V1^2}{2g} + Z1 = \frac{P2}{{rho}g} + \frac{V2^2}{2g} + Z2 + h_L
\frac{P1 - P2}{{rho}g} = \frac{V2^2 - V1^2}{2g} + Z2 - Z1 + h_L
Orifice meter
Q{thr} = \frac{A2}{\sqrt{\frac{1}{A2^2} - \frac{1}{A1^2}}} \sqrt{2g{\Delta}h}
Qa = Cd x Q_{thr}
{\Delta}h = (\frac{{{rho}m}}{{{rho}f}} - 1) x h_m
ΔP = ρ_f g Δh
If pressure difference given, find out the Δh from the following equation if manometer height difference is given.