## Fluid MechanicsProperties of Fluids Fluid Statics Control Volume Analysis, Integral Methods Applications of Integral Methods Potential Flow Theory Examples of Potential Flow Dimensional Analysis Introduction to Boundary Layers Viscous Flow in Pipes |
## Integral Approach
## Basic Concepts## VelocityFigure 1 : Velocity Field
The velocity
field defines a distribution of velocity in a given region, R
(
Velocity
can have three components, one in each direction,
Each of
Most
other variables involved in a fluid flow can also be given a field
representation. We have temperature field, ## Steady and Unsteady Flows
Velocity,
pressure and other properties of fluid flow can be functions of time.
If a flow is such that the properties at every point in the flow do
not depend upon time, it is called a For steady flows,
where
Any flow start up process is unsteady. Many examples can be given from everyday life: eg. water flow out of a tap which has just been opened. This flow is unsteady to start with, but with time can become steady.
Some flows, though unsteady, become steady by choosing a particular frame of
reference. These are called Unsteady flows are difficult to calculate while steady flows are less difficult having one degree less complexity. ## One, Two and Three Dimensional Flows
The
term Figure 2 : Example of one-dimensional flowConsider flow through a circular pipe. This flow is complex at the position
where the flow enters the pipe. But as we proceed downstream the flow
simplifies considerably and attains the state of a fully developed
flow. A characteristic of this flow is that the velocity becomes
invariant in the flow direction as shown in
Velocity at any location depends just on the radial distance
In comparison, flow through a diverging duct as shown in Figure
3: Example of a two-dimensional flow
The concept of a Figure 4 : Uniform Flow
## Flow Description, Streamline, Pathline, Streakline and TimelineStreamline, pathline, streakline and timeline form convenient tools to describe a flow and visualise it. They are defined below. Figure 5 : Streamlines Figure 6: Streamline definitionA streamline is a line that is tangential to the velocity vector at every point in the flow at a given instant of time. This definition leads to the equation for streamlines.
where Figure 7 : Streamtube
Due to its definition, a streamline has no
flow across it; i.e. there is no flow normal to a streamline.
Sometimes a bundle of streamlines is considered together inside a
general flow for analysis. Such a bundle is called Figure 8: PathlinesFigure 9: StreaklinesFigure 10: Timeline
In a steady flow the streamline, pathline and streakline all coincide. In an unsteady flow they can be different. Streamlines are easily generated mathematically while pathline and streaklines are obtained through experiments. ## Eulerian and Lagrangian approaches
Eulerian
and Lagrangian approaches are the two methods to study fluid motion.
The ## System and Control VolumeFigure 12 : Piston cylinder arrangement
The
term Figure 13 : Complex System Approach
It
is easy to analyse the system in the example of piston-cylinder
arrangement, but in fluid dynamics there are systems which are far
more complicated. For example, the flow about a moving vehicle. If a
system is defined by taking a volume of air at time Fig.
13. The boundary changes rapidly and undergoes unmanageable
distortions. The system approach is extremely difficult. Other
complex examples are flow through turbomachinery, flow in hydraulic
systems, etc.
Another
method is the Figure 14 : Control Volume, $ʊ$
Most
problems consider the control volume to be fixed, but it is possible
to have control volumes that change their boundary and deform. These
lead to more complicated solution equations. Examples of such control
volumes are shown in Figure 15 : Moving and Collapsible Control Volumes
The boundary of the control volume is referred to as The system and control volume approaches are akin to Lagrangian and Euler approaches respectively. ## Differential and Integral ApproachsThe
Figure 16: Differential and Integral approaches to calculate flow about an
Aerofoil.
It is not always necessary to get a detailed set of information of the
flow. A large control volume can be created to encompass the region ## Integral Equations## Basic Laws for Fluid FlowThere are three simple laws governing fluid flow. They are the conservation laws for mass, momentum and energy. All fluid systems are covered by these laws. ## Conservation of Mass
Consider a system of a fixed mass, Figure 17: A System
If mass is constant, then it does not change over time, so that, ## Newton's Second Law of Motion
Newton's
second law is the next one to be imposed upon fluid motion. It is
known that the rate of change of momentum is proportional to the
applied force. If
where
Momentum
This
law is also true for angular momentum $$ H = ∫_{system} (r × V) .dm = ∫_{system} (r × V)ρ.dʊ $$
which again is a vector equation. Torque ## Conservation of EnergyThe first law of thermodynamics which is a statement of the conservation of energy principle states, $$\text"i.e. "{dQ}/{dt}-{dW}/{dt}={dE}/{dt}$$
where In addition, Energy ( ## Second Law of Thermodynamics
While the first law of thermodynamics states that energy is conserved, the
second law establishes a direction in which a process can take place.
If
In addition to the above relations we may need an equation of state, ## Reynolds Transport Theorem
The
method of applying these conservation laws in a control volume approach is
called the
Starting with a system and the rate at which an extensive property ## Demonstration of the theorem for one-dimensional flow
Consider a stream tube in an one-dimensional flow. The flow takes
place entirely through the stream tube and there is no flow across
it, i.e., no flow in a direction normal to the stream tube. Consider the system I+II
while the system has moved to occupy the new position II + III. During the time
interval mass contained in region I
has entered the control volume and mass in III
has left the control volume.
Figure 18: Control Volume and system for an one-dimensional flow
Consider an extensive property
where subscript $$N_s=(N_{cv}-N_{I}+N_{III}) \text" at "t=t_0+Δt$$
On
substituting these into the above and noting that at By readjusting the terms, Each of the three limits on the RHS of the above equation can be simplified. The first limit gives,
where
The
second limit, which gives the rate of change of
The
right hand side is simply the rate at which
where
Similarly
we have for Upon substituting, This is the Reynolds Transport equation for the control volume considered. It represents
The above result can be generalised to any control volume of any shape,
but fixed in space. A general control volume is shown in
Figure 19: General Control Volume and System
Whether
the flow at any small area of control surface is an inflow or an
outflow is decided by the direction of the velocity vector and that
of the area vector at that segment. For a small area $dA$ at the control surface (
Integrating
this for the entire control surface gives the net rate of flow of Consequently, the Reynolds Transport theorem for a general control volume can be written as This equation can be simplified when specific control volumes are considered. ## Conservation of Mass
The
Reynolds Transport theorem can be applied to derive an equation for
conservation of mass. We note that in the equation,
Substitute
for By definition, a system is an entity of fixed mass, the left hand side of the above equation is zero, thus giving the equation for conservation of mass as
which
expresses that the rate of accumulation of mass within a control
volume is equal to the net rate of flow of mass into the control
volume. This equation is also called the ## Steady FlowFor a steady flow the time derivative in the equation vanishes. As a result,
In
addition if the flow is incompressible, ## Incompressible Flow
The equation simplifies further when we consider an incompressible flow
where density
Dividing by density, The first term is the rate of change of volume within a control volume, which for a fixed control volume is zero by definition. This gives a simple form of the equation for the conservation of mass for the control volume as Thus for an incompressible flow the continuity equation is the same irrespective of whether the flow is steady or unsteady. ## $V↖{→}× dA$
This
where
A negative ## Application to a one-dimensional control volume
Consider an one-dimensional stream tube flow as shown in Figure
20 : Control Volume for an one-dimensional steady flow
If a uniform flow prevails at surfaces V
and areas
of cross section, _{2}A
and _{1}A, then the
application of continuity gives_{2}as flow properties are constant over the areas this simplifies to ## Momentum Equation
A momentum equation resulting from the Reynolds Transport theorem can
also be derived. Now, Consider the left hand side of Reynolds Transport equation. ${dM↖{→}}/{dt}$ is proportional to the applied force as per Newton's Second Law of motion. Thus,
Where
Now
substitute for Writing this as three equations, one for each coordinate direction,
The
term $uρV↖{→}×dA$ represents
the As stated before the term $ρV↖{→}×dA$ is replaced by $ρV\cos(α).dA$ to account for the angle between the normal to the surface and the direction of the flow. This equation set is used in numerous applications in fluid dynamics, such as force at the bending of a pipe, thrust developed at the foundation of a rocket nozzle, drag about an immersed body etc. ## Bernoulli Equation
The momentum equation leads to the development of the
A differential stream tube within a flow with a defined small control volume
within it is shown in ds.
Figure 21: Differential Control Volume for an one-dimensional steady flow
Since it is a small stream tube any property changes only vary slightly
in direction P. Assuming the flow is incompressible, then at outlet end (2), the corresponding properties will be
A+dA, V and _{s}+dV_{s}, ρ+dρP+dP.Applying the momentum equation to the above differential control volume will provide a detailed solution to the problem. First a list of the assumptions made is required.
- A stream tube with no cross flow considered.
- The flow is steady, ${∂}/{∂t}=0$
- Fluid is incompressible (
**ρ**= constant,**dρ = 0**).
Any application of the momentum equation should be preceded by the Continuity Equation. Complete information about the flow cannot be obtained by applying the momentum equation alone. ## Application of Continuity Equation
The first term in the equation cancels out because of the steady flow
assumption. Since all the flow takes place through surfaces ( giving where ${dm}/{dt}$ is the mass flow rate through the control volume. ## Application of Momentum EquationFrom the momentum equation,
Since
the flow is steady, the first term on the RHS drops out. Body forces F
acting on the control volume need to be evaluated.
_{Ss}## Body ForcesThe only body force acting is the weight of the fluid within the control
volume. Accordingly, in the ## Surface Forces
The surface force is due to pressure acting upon the boundaries of the
control surface. There are three terms that contribute - end ( ## Terms from the Right Hand Side integrations
substituting for Now equating terms from the LHS and RHS, i.e., i.e.,
The above equation is readily integrated for an incompressible flow (
This equation is called the Bernoulli Equation. Note that it connects
pressure ( This equation is valid for steady flows only in absence of any friction such as forces due to viscosity and the flow must be incompressible. ## Application to moving Control Volumes
The continuity and the momentum equations can be extended to cases where
the control volume is not fixed in space. One such case is when the
control volume is moving with a constant velocity, say an aircraft
or a ship moving at a constant speed. Note that the equations derived
assume that the speeds are all referred to the control volume. So it
becomes a simple matter to consider a control volume moving at a
constant speed, which now is the speed relative to the control volume. The equation for Reynolds Transport theorem, gets altered to ## Equation for Angular MomentumMany flow devices and turbo-machinery involve rotating components. Examples are Centrifugal pumps, Turbines and Compressors. The analysis of such systems is facilitated by the Reynolds Transport theorem written for angular momentum. where,
It becomes necessary now to calculate the angular momentum about some
point, say Substitution into the equation for Reynolds theorem gives,
The LHS of the above equation is the sum of all the moments about the
point ## Deformable Control Volumes and Control Volumes with non-inertial accelerationIt is possible to extend this analysis to the general cases of deformable control volumes and those that undergo acceleration, but these are beyond the scope of this text. ## Energy EquationThe Reynolds Transport theorem can also be used to derive an equation for energy conservation in a control volume. Now, On the LHS we have ${dE}/{dt}$, which from the First Law of Thermodynamics is Where ${dQ}/{dt}$ is the rate at which heat is added to the system and ${dW}/{dt}$ is the rate at which work is done on/by the system. Substituting,
In the above equation, For simple fluids,
There are different modes of performing work - shaft work,
where
Work due to shear forces is small and is usually neglected. Heat added ${dQ}/{dt}$ becomes important only in problems involving heat transfer. Upon substituting for various terms,
where
## Energy equation for a one-dimensional control volumeFigure 22 : Control Volume for a one-dimensional steady flow
Consider the one-dimensional control volume shown in Note that for continuity, $(ρAV)_1=(ρAV)_2={dm}/{dt}$
On division by ${dm}/{dt}$
and denoting ${{dQ}/{dt}}\/{{dm}/{dt}}$ by
Note that the term $(h+gz+1/2V^2)$ is equal to the Total Enthalpy denoted by The total enthalpy of a control volume is conserved unless heat or work is added to or taken out of the control volume. ## Low Speed Application
In low speed applications, it is usual to express energy as a or
The term $P/γ$ is
called the If we consider a simple pipe flow without the shaft work then the equation becomes
The terms within the parenthesis are called the The losses that take place between "inlet" i.e., (1) and "outlet" i.e., (2) are obtained through measurements and imperical correlations. ## Relationship between Energy Equation and Bernoulli EquationAn examination of these equations brings out the connection between the Energy equation and the Bernoulli equation. It is clear that two equations become one when losses that occur between (1) and (2) are ignored. Bernoulli Equation can be used only when considering a frictionless flow along a streamline. Further it is required that the flow be incompressible without any addition of heat or shaft work. ## Bernoulli Equation for Aerodynamic FlowIn aerodynamics, considerably higher speeds are dealt with. An aircraft flies at speeds of the order of 500 kmph and more, while river flows or household pipe flows may only involve 10 kmph or so. Consequently, the kinetic energy term for aerodynamic flows is very large when compared to the potential energy. Accordingly, it is usual to neglect potential energy for such flows. The Bernoulli Equation as a consequence becomes, ## Stagnation PressureFigure 23 : Stagnation Point on (a) Simple Body and (b) a complicated Body
The Bernoulli equation can be applied to flow about a body such as an
aircraft as shown in
where
The
term " ## Energy Grade Line
Terms
where
term Figure 24: Energy Grade Line (EGL) and Hydraulic Grade Line (HGL) for an
one-dimensional flow.
If the losses are taken into account the EGL will drop accordingly. Any work extraction along the path, such as via a turbine, will be seen as a sudden drop in the EGL. Any work addition will be reflected as a sharp rise. HGL follows similar trends. ## Kinetic Energy Correction FactorIt is assumed in the derivation of Bernoulli equation that the velocity at the end sections (1) and (2) is uniform, but in practical situations this may not be the case and the velocity can vary across the inlet or exit cross sections. A solution for handling this additional complexity is to use a correction factor for the kinetic energy term in the equation. If $V_{avg}$ is the average velocity at an end section then, for energy, After simplification, Consequently,
where |