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PPT – AerodynamicsB, AE2115 I, Chapter II PowerPoint presentation | free to view - id: 125516-MmFjO

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Aerodynamics Some Fundamental Principles and

Equations

- Dr.ir. M.I. Gerritsma Dr.ir. B.W. van

Oudheusden - Delft University of Technology
- Department of Aerospace Engineering
- Section Aerodynamics

Continuity Equation, Differential Form

- We already found that

Remember that V and V are fixed in space,

therefore

Apply divergence theorem

So the continuity equation becomes

Continuity Equation, Differential Form

- This equation holds for any arbitrary volume V

If this holds for every imaginable volume V, then

the integrand must be zero, so

Validity 3D, time-dependent, viscous-inviscid,

(in)compressible, allows for discontinuous

solutions (!?)

Stationary flows

Continuity Equation, Differential Form

- Conservative formulation
- Non-conservative formulation

Physical Meaning of the Divergence of Velocity

- Moving control volume
- Constant mass
- Shape changes

Physical Meaning of the Divergence of Velocity

- Assume that the volume VV is so small that

is constant in V, then

is physically the time rate of change of

the volume of a moving fluid element, per unit

volume

Momentum Equation

- Newtons second Law

Physical Principle Force time rate of change

of momentum.

Note and are vectors, so the above

identity has to hold for each component.

- 1. Body Forces
- Gravity
- Electromagnetic forces
- Coriolis forces

Which types of forces act on a fluid?

- 2. Surface forces
- normal stresses (perpendicular to plane)
- tangential forces (along the plane)

Body Forces

Fixed control volume V

Assume is the body force per unit mass

exerted on fluid inside V

The body force acting on a small volume dV

Total force on volume

Example, gravity

Pressure contribution

- Consider a small elementary surface
- On this surface the pressure p acts, therefore
- in the opposite direction of the outward unit

normal - Therefore the total pressure force on V equals

Viscous forces

- The viscous forces are simply denoted by

. In Aerodynamics-D AE3-130 explicit formulae for

will be given.

We want to find an expression for

- Consists of
- Body forces
- Pressure forces
- Viscous forces

Therefore

- We want to find an expression for

Question How do we evaluate the rate of change

of momentum in the volume V?

- Answer 2 contributions G and H
- G Net flow of momentum out of control volume

across surface S. - H Time rate of change of momentum due to

unsteady fluctuations of flow properties inside V

G Recall that the mass flow across the elemental

area ds is given by Hence, the flow of momentum

per second across ds is Integrating over the

whole surface yields Note that when Ggt0 momentum

is flowing out of the control volume.

- H Time rate of change of momentum due to

unstaedy phenomena in V.

Question What is the total amount of momentum in

V at time t.

Answer Within a small volume element dV with

velocity and mass

Therefore the time rate of change of momentum in

the volume V is given by

Integral form of the momentum equation

- Newton

Therefore

Body forces

Pressure forces

Viscous forces

Momentum transport across surface

Rate of change of momentum in V

Some Remarks concerning the Momentum Equation

- The momentum equation is a vector equation. In

fact we have three separate equations in the

x-direction, y-direction and z-direction. - The integral equation is valid for unsteady,

viscous, (in)compressible, three dimensional

flows. - will be treated separately

(Aerodynamics-D AE3-130) - The integral formulation only provides

information over integrated quantities, so no

local values of density, pressure, temperature,

etc, can be extracted.

Conversion to differential form

- Recipe
- Convert all surface integrals to volume

integrals using the integral relations given in

section 2.2.11. - Interchange and which is allowed

since V is fixed in space. - Collect all integrands underneath one integral

and require that the integrand must vanish, since

the integral holds for arbitrary volumes.

- The momentum equation contains the surface

integral - Apply the gradient theorem

In order to rewrite the convective surface

integral, we look at the x-component only

Take the x-component of the momentum equation

- Apply divergence theorem to remaining surface

integral (Convective term)

Writing the viscous force as

.

Additional Remarks

- These equations are called the Navier-Stokes

equations. They consist of a system of non-linear

partial differential equations (PDEs). - The PDE connect the variables and the change in

the variables locally. - If the flow is stationary,

inviscid and there are no body

forces we obtain the so-called Stationary Euler

equations

Energy Equation

- We have introduced now 5 unknowns,

and we have only 4 equations, namely

the three components of the momentum equation and

the continuity equation. - We either have to add another equation, or remove

one of the unknowns. - If the flow is incompressible, then the density

is constant, so we can remove one unknown which

closes the system - For compressible flows, one cannot remove any

unknowns, therefore one has to add another

equation. This will be the energy equation. - Physical Principle Energy cannot be created, nor

destroyed.

First Law of Thermodynamics

Increment in internal energy

Work done on the system

Added heat

Energy Equation.

- Apply to a control

volume

- Call
- B1 the amount of heat added to the volume from

the environment - B2 the amount of work exerted on the volume
- B3 rate of change of energy as it flows

through the control volume

B1 B2 B3

Determination of added heat B1

- A small volume element dV receives from its

environment an amount of heat per unit time and

unit mass

So the total amount of heat added to dV per unit

time is

Sources are radiation, emission, combustion,

condensation, etc.

- Through the surface V heat is added to the

volume by conduction and mass diffusion. These

are so-called viscous processes, denoted by

So the total heat added to the system, B1, will

be equal to

Rate of work done on the volume, B2

- Work

- On surface element a pressure force

acts, so the amount of work per unit

time due to the pressure will be

So the

total pressure contribution to the rate of work

over the entire volume is given by - On dV a body force acts, therefore

the rate of work due to the body force on dV is

The total rate of work on V is

given by

- Viscous rate of work

Determination of the rate of change of energy, B3

- Energy in V consists of internal energy, e per

unit mass and kinetic energy per

unit mass, so the total energy per unit mass is

given by

At time t the amount of energy in dV is given by

So the instationary rate of change of energy

- The rate of mass flowing through the surface

carries a net amount of energy given by

Energy Equation

Thermal conduction, mass diffusion

Work done by body forces

Work done by pressure forces

Volumetric heating

Work done by viscous forces

Net rate of energy outflow

Transient fluctuations

- This is the integral form of conservation of

energy, valid for (in)compressible,

viscous/inviscid fluids, stationary and time

dependent problems. - Introduction of the enthalpie
- Introduction of the total enthalpie

Energy Equation in terms of H

- Total energy
- total enthalpy

Differential Formulation of Energy Equation

- Divergence Theorem

- Results

Special Cases Energy Equation

- If the flow is
- stationary
- inviscid
- adiabatic
- no body forces

Conclusion The change in H in the direction of

the flow is zero (remember directional

derivative), therefore H does not change in the

direction of the flow for a stationary,

inviscid, adiabatic flow without body forces the

total enthalpy is constant along streamlines!

Important Concepts in Aerodynamics

- Substantial, Convective, Local Derivative,
- Pathlines, Streamlines and Streaklines,
- Angular Velocity, Vorticity and Strain
- Circulation
- Stream Function
- Velocity Potential
- Relationship between Stream Function and

Velocity Potential

section 2.9 2.10 section 2.11 section

2.12 section 2.13 section 2.14 section

2.15 section 2.16

Substantial Derivative

- When we consider a scalar quantity q we can

either write this quantity with respect to fixed

points in space, X, or with repect with a

coordinate system that is moving with the flow,

x. - Obviously, x, will be a function of time t, so

x(t). - In the first case, we have the following time

derivative - In the second case we have

Substantial Derivative

- So we have the new time derivative
- This can be written more succinctly as
- The new time derivative is called the substantial

or material time derivative

Convective Derivative

Substantial Derivative

Local Derivative

Fundamental Equations in terms of the Substantial

Derivative

- Conservation of Mass
- Conservation of Momentum
- Conservation of Energy

Pathlines Streamlines

- A pathline is the curve in space traced out by a

particular particle in time - A streamline is a curve in space which is

locally tangent to the velocity vector

For steady flows pathlines and streamline

coincide, for unsteady flows these lines are

generally different!

Direction of a Streamline

- Since the curve is locally tangent to the

velocity vector we have

Writing

So the equations for a streamline are

Vorticity Strain

- Different aspects of the movement of a particle
- The trajectory (already discussed)
- Orientation (translation and rotation)
- Deformation.

- Rotation and deformation are determined by the

velocity field. - Consider a 2 dimensional square

- Convention counter clockwise rotations are

positive - The angle is given by
- and the rotation of AB by

- Definition of angular velocity
- The angular velocity of a small line element is

given by - Therefore the angular velocity of AB is
- And the angular velocity of AC is
- The angular velocity of the fluid element ABCD

is given by the average angular velocity

- For general 3 dimensional flows the angular

velocity is a vector given by

Vorticity is defined as

Irrotational Flow

- If at every point in the flow,

the flow is called rotational. - If at every point in the flow,

the flow is called irrotational.

Irrotational Flow

- Irrotational flow is of practical use
- - Subsonic flow around airfoils
- - Supersonic flow around slender bodies (no

curved shock waves) - - Subsonic/Supersonic flow in nozzles, jets,

wind tunnels, etc.

Note, that in many flows the flow outside the

boundary layer may be assumed to be irrotational,

but inside the boundary layer the flow is

rotational, since

Strain / Deformation

- Strain is the change in k, where positive strain

corresponds with decrasing strain

Time rate of strain

- Definition Time rate of strain
- Deformation rates!
- Analogously

Circulation

- Circulation is strongly connected to the notion

of lift - Frederick Lanchaster (1878-1946)
- Wilhelm Kutta (1867-1944)
- Nicolai Joukowski (1847-1921)
- Definition

- Remarks
- Circulation depends on the contour C chosen
- It is customary to use a positive circulation in

the clockwise direction!!

Connection between Circulation and Vorticity

- The circulation of a flow around a closed curve C

is by definition - Now using Stokes Theorem we get

Circulation and Vorticity

- Stokes Theorem applied to circulation yields the

following result - The circulation around a contour C can also be

obtained by integrating the vorticity over the

surface enclosed by this contour - Remarks
- If everywhere (i.e. an

irrotational flow) the circulation is zero for

every contour C. - Conversely, if for every imaginable

contour C, then the vorticity must be zero

everywhere and therefore the flow will be

irrotational. - Suppose that the contour shrinks such that it

encloses a very small surface over which the

vorticity is more or less constant, then the

circulation is given by

Stream function

- Consider a two dimensional flow
- The definition of a streamline is given by

This equation constitutes a differential equation

with solution in which denotes the

integration constant. Each different constant

defines a different streamline. Suppose that

streamline A corresponds to the constant

and streamline B corresponds to

is called a stream function if

denotes the mass flow between the

streamlines A and B.

Stream function

- Mass flow

Remark Differentiating the stream function in

the direction perpendicular to the streamline

produces the mass flux

Stream function

- If the streamfunction is given, the above

formulae give a way to determine the momentum per

unit volume - Note that differentiation of the stream function

in the y-direction produces the mass flux in the

x-direction and vice versa. - For an incompressible flow we can redefine the

streamfunction by setting by dividing by the

constant density, in which case the above

formulae produce the velocity components. - In polar coordinates the above equation become
- The stream function provides a very usefull

tool given the stream function we can

immediately determine the streamlines by setting

yconst and we can determine the velocity field - Note that the velocity components obtained from

te stream function identically satisfy the

stationary continuity equation.

Velocity Potential

- The introduction of a velocity potential only

makes sense for irrotational flows. - Irrotational
- Vector identity
- Therefore set

Potential Function

- Remarks
- If is given, the velocity

components follow immediately by differentiation,

so the potential function replace 3 unknowns u, v

and w, by 1 unknown - The velocity component in a particular direction

is obtained by differentiating the potential

function in that particular direction. - By introducing the potential function we

identically satisfy the condition for

irrotational flow, i.e. - By replacing the velocity components by the

gradient of the potential function in the

continuity equation and momentum equation we

obtain the so-called potential equation.

Irrotational flows satisfying this equation are

called potential flows.

Relation between the stream function and the

potential function

- Consider a two dimensional, incompressible flow
- Continuity equation
- Streamlines
- Equipotential lines

We know that is orthogonal to

the equipotential lines and oriented in the

direction of the flow so are the streamlines,

therefore equipotential lines are orthogonal to

the streamlines. A mathematical proof of this

assertion can be found in section 2.16.

Proof orthogonality of streamlines and

equipotential lines

- Note that is perpendicular to the

streamlines and is perpendicular to the

equipotential lines, in which - so these gradients are perpendicular to

eachother, and therefore the lines whose normals

they are, are perpendicular as well.

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