MECH1907
Introduction to Aerospace Engineering
Rhea P. Liem
rpliem@ust.hk
Fluid Dynamics
September 12th and 17th , 2019
Recap: Fundamental Thoughts
1
Four fundamental quantities in aerodynamics: pressure p, density ρ,
temperature T , and velocity V or ~v . Streamlines are typically used to
illustrate the flow field.
2
Flow field around the body → pressure and shear stresses → aerodynamic
forces
3
Perfect gas: gas where the intermolecular forces between particles are
negligible.
4
Equation of state
p = ρRT
or
pv = RT ,
where R is the specific gas constant.
5
It’s important to use consistent units! In this course, we use SI and
English units.
1
Learning objectives
1
Relate the change in pressure dp and change in vertical height dy when
the fluid is in equilibrium
2
Compute the vertical pressure force acting on a body immersed in a fluid
3
Understand the different types of flows
4
Understand the concept of boundary layers
5
Understand the 3 fundamental physical laws for aerodynamics
6
Understand why the nozzles of rocket engines, or garden hoses, are shaped
the way they are
7
Understand the relation between pressure and velocity along the streamline
8
Understand the relation between temperature and velocity
2
Four pillars in the design of a flight vehicle
1
Aerodynamics
2
Flight dynamics
3
Propulsion
4
Structures
3
Continuum mechanics
Continuum Mechanics
Fluid Mechanics
Fluid Statics
Solid Mechanics
Fluid Dynamics
Aerodynamics
Hydrodynamics
4
Continuum mechanics
Continuum Mechanics
Fluid Mechanics
Fluid Statics
Solid Mechanics
Fluid Dynamics
Aerodynamics
Hydrodynamics
4
Introductory Concepts and Definitions
Fluid: liquid or gas
Fluid statics: the study of fluid at rest
Fluid dynamics: the study of the effect of forces on fluid motion
We will focus mainly on aerodynamics.
Hydrostatics and Hydrodynamics are usually used for situations
involving liquids.
Aero- and Hydro- disciplines are actually similar, but with different
applications (e.g., airplanes vs ships).
5
Aerostatics and Aerodynamics (Merriam Webster)
Aerostatics
A branch of statics that deals with the equilibrium of gaseous fluids and of solid bodies
immersed in them.
Fatm
Fatm
Wgas = ρgas V g
Aerodynamics
A branch of dynamics that deals with the motion of air and other gaseous fluids, and
with the forces acting on bodies in motion relative to such fluids.
6
Fluid dynamics in modern engineering devices
Airplanes
Ship
Automobiles
Wind turbines
Biomedical
7
Brief History of the Early Development of Fluid Dynamics
Aristotle (384–322 B.C.)
The concept of continuum
The concept of fluid dynamic drag, i.e., a resistance acting on the body
Archimedes (287–212 B.C.)
The concept of pressure in fluid statics
The concept of pressure gradient in fluid dynamics: a difference in pressure
must be exerted across the fluid to set a stagnant fluid into motion
Leonardo da Vinci (1452–1519)
The continuity equation, AV = constant
Edme Mariotte (1620–1684) and Christian Huygens (1629–1695)
The velocity-squared law, F ∝ V 2
Isaac Newton (1642–1727)
The concept of drag (resistance when a solid body moving through a fluid) and
viscosity (Newtonian shear-stress law)
8
Brief History of the Early Development of Fluid Dynamics
(cont’d)
Daniel Bernoulli (1700-1782)
The pressure-velocity concept: pressure decreases as velocity increases
Leonhard Euler (1707–1783)
The governing equations of inviscid fluid motion (Euler equations)
Louise Marie Henri Navier (1785–1836) and George Gabriel Stokes
(1819–1903)
Inclusion of friction in theoretical fluid dynamics (Navier-Stokes equations)
Osborne Reynolds (1842–1912)
Understanding turbulent flows (Reynolds number)
Wilhelm Kutta (1867–1944) and Nikolai Joukowsky (Zhukovsky) (1847–1921)
The circulation theory of lift (Kutta-Joukowsky theorem)
Ludwig Prandtl (1875–1953)
Boundary-layer theory
9
Fluid Statics: Buoyancy Force
Hydrostatic equation
A differential equation which relates the change in pressure dp in a fluid with a
change in vertical height dy , for a stationary fluid element (in equilibrium).
Buoyancy Force
Vertical pressure force acting on an arbitrary body immersed in a fluid
See derivations on the board!
10
U-tube Manometer
Used to measure the pressure difference ∆p, e.g., in wind-tunnel experiments.
Find the force balance at plane B–B:
p1 A = p2 A + ρgA∆h
And therefore:
p1 − p2 = ρg ∆h
11
Example – Fluid Statics and Buoyancy
A hot-air balloon with an inflated diameter of 30 ft is carrying a weight of 800
lb, which includes the weight of the hot air inside the balloon.
Calculate:
1
Its upward acceleration at sea-level the instant the restraining ropes are
released
2
The altitude reached where the net lift force is 0
Assume that the variation of density in the standard atmosphere is given by
4.21
ρ = 0.002377 1 − 7 × 10−6 h
, where h is the altitude in ft and ρ is in
slug/ft3 . Use the gravitational acceleration g = 32.2 ft/s2 .
12
Block diagram categorizing the types of aerodynamic flows
Aerodynamics
Continuum Flow
Free-molecular flow
Viscous flow
Inviscid flow
Incompressible flow
Compressible flow
Subsonic flow
Transonic flow
Supersonic flow
Hypersonic flow
13
What Do We Need to Derive Aerodynamic Equations?
Basic philosophical procedures:
1
Invoke three fundamentals physical laws:
Conservation of mass
Mass can be neither created nor destroyed
Conservation of momentum
Newton’s second law: force = mass × acceleration
Conservation of energy
Energy is conserved; it can only change from one form to another
2
Determine the suitable model of the fluid
→ compressible/incompressible, viscous/inviscid, ...
3
Apply the fundamental physical principles in (1) to the model of fluid in (2)
14
Continuum vs molecular description of fluid
Liquid and gases are made up of molecules in random motion. The molecules
are constantly in contact with each other, so at macroscopic scales they act as a
uniform fluid material.
Mean free path
Definition: the average distance the molecule travels before colliding with each
other. For air:
Mean free path at 0 km: 0.0001 mm
Mean free path at 20 km (Lockheed U-2 flight): 0.001 mm
Mean free path at 50 km (balloons): 0.1 mm
Mean free path at 150 km (low orbit): 1000 mm = 1 m
15
Continuum and free molecular flow
Continuum flow
When the mean free path vehicle dimension, the flow appears to the body as
a continuous substance. The molecules impact the body surface so frequently
that the body cannot distinguish the individual molecular collisions.
Examples: any atmospheric vehicles (airplanes, helicopters, etc)
Free molecular flow
When the mean free path ≈ vehicle dimension, the gas molecules are spaced so
far apart that collisions with the body surface occur only infrequently → the
body surface can feel distincly each molecular impact.
Examples: orbital satellites, space shuttle at the extreme outer edge of the
atmosphere
16
Viscous and Inviscid Flows
Viscous Flow
A flow that leads to some “transport phenomena”: mass diffusion, viscosity (friction),
and thermal conduction
Inviscid Flow
A flow that involves no friction, thermal conduction, or diffusion.
Reynolds number Re → ∞, or a very large number.
17
Frictionless Flow
Due to the lack of friction, the streamline right at the surface slips over the
surface.
This flow is unrealistic!
18
Boundary Layer Concept – in Real Flows
There is friction between the gas and the solid material
The flow at the surface adheres to the surface
There is a thin region of the retarded flow: boundary layer
The flow velocity is zero at the surface
19
Boundary Layer Effects (Introduced by Ludwig Prandtl)
A flow field can be split into two regions:
1
Boundary layer near the surface (where friction is important)
2
Frictionless flow (or potential flow) outside
Wall shear stress, τw
Depends on the absolute viscosity coefficient, or viscosity (µ), and the velocity
gradient at the wall (( dV / dy )y =0 ).
τw = µ
dV
dy
y =0
For air at the standard sea-level temperature, µ = 1.7894 × 105 kg/(m)(s).
20
Reynolds Number
Growth of the boundary layer thickness
Let x be measured from the leading edge, and V∞ be the flow velocity far
upstream of the plate, or the free-stream velocity.
The local Reynolds number (as a function of x) is thus:
REx =
ρ∞ V∞ x
µ∞
Re is a dimensionless quantity.
21
Laminar and Turbulent Flows
Laminar flow
The streamlines are smooth and regular
Turbulent flow
The streamlines break up and a fluid element moves irregularly and randomly
22
Incompressible and Compressible Flows
Incompressible Flow
A flow in which the density ρ is constant
Compressible Flow
A flow where the density ρ is variable
Illustration of compressibility
23
Incompressible flow assumption
No flows in nature are truly incompressible (where ρ is precisely constant), but
some can be modeled as incompressible:
The flow of homogeneous liquids (in hydrodynamics)
Gases at low Mach number, M < 0.3
(Mach = velocity/speed of sound)
e.g., early aircraft (from Wright brothers’ to before World War II) and some
small aircraft today
24
Flow Classification based on Mach Number
The Mach number is defined as the ratio of the freestream speed to the speed of sound
M∞ ≡
V∞
a∞
The speed of sound a∞ is the velocity at which pressure disturbances travel in the fluid.
Subsonic flow (M < 1 everywhere)
Characterized by smooth streamlines
E.g., Wright brothers’ plane, small aircraft
Transonic flow (mixed regions where M < 1 and M > 1)
If M∞ is subsonic but is near unity, the flow can be locally supersonic (M > 1).
For slender bodies, 0.8 < M∞ < 1.2. E.g., most transport aircraft nowadays
Supersonic flow (M > 1 everywhere)
Characterized by the presence of shock waves (flow properties and streamlines
change discontinuously)
E.g., Concorde (discontinued in 2003)
Hypersonic flow (very high supersonic speeds, M > 5)
E.g., rockets
25
Before we derive the laws...
Recall the three laws...
1
Conservation of mass
2
Conservation of momentum
3
Conservation of energy
... we first need to define the specific quantity of fluid we are dealing with.
→ control volume.
26
Control Volumes to Model Fluid Elements
Treat fluid elements as a continuous deformable medium instead of isolated point
masses. Using control volumes, which could be:
Finite: a closed volume drawn within a finite region of the flow (control
volume V). Control surface S refers to the closed surface which bounds the
control volume
Infinitesimal: infinitesimally small fluid element in the flow, with a
differential volume dV
Two types of control volumes can be employed:
Eulerian type: volume fixed in space, and fluid can freely pass through the
volume’s boundary
Lagrangian type: volume is attached to the fluid. Volume is freely carried
along with the fluid, and no fluid passes through the boundary.
27
Conservation of mass – derivation of the general equation
(For your background information.)
Physical principle: mass can never be created nor destroyed
Applied to a finite fixed control volume, with the possibility of mass flow across the
volume boundary
Rate of change of mass in volume = mass flow into the volume
d
dt
˚
‹
ρ dV = −
~ · n̂ dS
ρV
How to interpret this equation?
28
Conservation of mass derivation (get the intuition right...)
d
dt
˚
d
Left hand side (LHS) term:
dt
‹
ρ dV = −
~ · n̂ dS
ρV
˚
ρ dV
˚
Triple integration
= volume integration (in a 3-dimensional space)
‹
Right hand side (RHS) term: −
~ · n̂ dS
ρV
¨
Double integration
= surface integration (in a 2-dimensional space)
˛
Cyclic integration
= for a closed surface
~ · n̂ = the dot (scalar) product between the velocity vector and the normal
V
~ · n̂ = V
~ |n̂| cos θ.
vector from a point on the surface, V
29
Maths: relationship between a volume integral and a closed-surface
integral (additional info)
We will use the Divergence Theorem (also called the Gauss’ Theorem):
˚ ‹
~ dV
~ · n̂ dS =
∇·V
V
~
Divergence of a vector field, ∇ · V
Like gradient information, it gives us a measure of how much the vector field is
“spreading out” at each point.
~ = ∂Vx + ∂Vy + ∂Vz
∇·V
∂x
∂y
∂z
Positive divergence
Negative divergence
Zero divergence
30
The continuity equation in the form of partial differential
equation (additional info)
Applying the Gauss’ Theorem on the conservation of mass:
˚ dρ
~
+ ∇ · ρV
dV = 0
dt
This equality must be valid to all control volumes, even at the infinitesimal level dV:
dρ
~ =0
+ ∇ · ρV
dt
What is the steady flow version of this equation?
What is the steady incompressible flow version of this equation?
31
Conservation of mass on a steady unidirectional channel
flow (stream tube)
There is no flow across streamlines
In moving from point 1 to 2, the cross-sectional area of the tube might change
For a steady flow, the mass that flows through the cross section at point 1 and 2
must be the same – “What goes in one end must come out the other end.”
Continuity equation → ρ1 A1 V1 = ρ2 A2 V2
Before we derive it, let’s see what are mass flow and mass flux.
32
The flow of fluid across a fixed control volume
Consider a small patch of the surface of the fixed control volume, with area A
and normal unit vector n̂.
Swept volume
The volume swept by the plane of fluid particles on the surface between time t and
t + ∆t
~ · n̂
∆V = Vn A ∆t, where Vn = V
Mass flow and mass flux
Mass flow, ṁ = ρ Vn A
time rate of mass passing through the area
Mass flux = ρ Vn
mass flow per area
See derivations on the board!
33
Continuity equation for streamlines of flow over an airfoil
The stream tube does not have to be bounded by a solid wall
Consider the streamlines of flow over an airfoil, the space between two adjacent
streamlines is a stream tube
34
Conservation of momentum
An equation relating pressure and velocity based on the Newton’s second law
Forces acting on a fluid element
Pressure forces acting normal to the element
Frictional shear forces acting tangentially on the surface
Gravitational force acting on the mass inside the element
Euler’s equation/Momentum equation
Applying the Newton’s second law, F = ma, to an inviscid fluid element with negligible
gravitational force
dp = −ρV dV
See derivations on the board!
35
Deriving the Euler’s/momentum equation
Consider an infinitesimally small fluid element (at point P) moving along a streamline
with velocity V .
Ignoring the frictional and gravitational forces, define the expression for F
Define the mass of the fluid element m
Define the acceleration of the fluid element a
Apply the Newton’s second law, F = ma
36
Bernoulli’s Equation
We want to relate the pressure and velocity at 2 different points along the streamline...
By integrating the Euler’s equation and assuming an incompressible flow:
p1 + ρ
V12
V2
= p2 + ρ 2
2
2
Bernoulli’s equation
p+ρ
V2
= constant along the streamline
2
See derivations on the board!
37
Example – Conservations of Mass and Momentum
Consider a convergent duct with an inlet area A1 = 5m2 . Air enters this duct
with a velocity V1 = 10m/s and leaves the duct exit with a velocity
V2 = 30m/s. Air pressure and temperature at the inlet are p1 = 1.2 × 105 N/m2
and T1 = 330K . Assume an incompressible flow.
Compute:
1
What is the area of the duct exit?
2
What is the pressure at the exit?
38
Example – Venturi Tube
Consider a venturi with a small hole drilled in the side of the throat. This hole is
connected via a tube to a closed reservoir. The venturi has a throat-to-inlet area
ratio of 0.85.
Calculate the pressure difference between the inlet and throat when the venturi is
placed in an airstream of 90 m/s at standard sea level conditions. Assume an
inviscid and incompressible flow.
For standard sea level:
Density: 1.23 kg/m3 (0.002377 slug/ft2 )
Pressure: 1.01 × 105 N/m2 (2116 lb/ft2 )
39
Basic Thermodynamics
In studying compressible flows, we need to examine the consequences of energy
changes in a gas on the flow’s pressure and temperature.
First Law of Thermodynamics
The change in internal energy de is equal to the sum of the heat added to δq and the
work done on the system δw .
δq + δw = de
40
Relations with the Fundamental Quantities
1
First, we express the work done as a function of p and v
2
Define enthalpy
h = e + pv = e + RT
3
Two important equations (we will use them again and again!)
δq = de + p dv
δq = dh − v dp
See derivations on the board!
41
Constant Volume and Constant Pressure Process
Constant-volume process: δq → dp, dT
Constant-pressure process: δq → dT , dv (and hence dρ)
42
Specific Heat
By definition, specific heat is the heat added per unit change in temperature of
the system
δq
c≡
dT
For constant-volume and constant-pressure processes:
δq
δq
cv ≡
and cp ≡
dT constant volume
dT constant pressure
We can then derive 4 important equations, relating thermodynamics variables
only (e, h, T )
de = cv dT
e = cv T
→
dh = cp dT
→
h = cp T
See derivations on the board!
43
Isentropic Flow
Definitions
An adiabatic process: no heat is added or taken away, δq = 0
A reversible process: no frictional or other dissipative effects occur
An isentropic process: both adiabatic and reversible
Most aerodynamic flows (e.g., the flow of air over the airfoil, flows through wind
tunnel nozzles and rocket engines) can be assumed to be isentropic.
Using some equations we have obtained before, we can derive the relations
between p, ρ, and T between two different points on a streamline in an
isentropic flow:
γ γ/(γ−1)
p2
ρ2
T2
=
=
p1
ρ1
T1
where γ ≡ cp /cv (= 1.4 for air).
See derivations on the board!
44
Energy Equation
Physical principle: Energy can be neither created nor destroyed.
It can only change form.
In other words: energy is conserved.
From the first law of thermodynamics, we can obtain:
h1 +
V12
V2
= h2 + 2
2
2
→
h+
V2
= constant
2
Using specific heat, we can relate the temperature and velocity at two different
points along the streamline:
cp T1 +
V12
V2
= cp T2 + 2
2
2
→
cp T +
V2
= constant
2
See derivations on the board!
45
Example – Basic Thermodynamics and Energy Equation
Consider a supersonic wind tunnel:
The air temperature and pressure in the reservoir of the wind tunnel are
T0 = 1000K and p0 = 10 atm, respectively. The static temperatures at the
throat and exit are T ∗ = 833K and Te = 300K. The mass flow through the
nozzle is 0.5 kg/s. For air, cp = 1008J/(kg)(K).
Calculate:
1
Velocity at the throat V ∗
2
Velocity at the exit Ve
3
Area of the throat A∗
4
Area of the exit Ae
46
Learning objectives
1
Relate the change in pressure dp and change in vertical height dy when
the fluid is in equilibrium
Hydrostatic equation
2
Compute the vertical pressure force acting on a body immersed in a fluid
Buoyancy
3
Understand the different types of flows
Inviscid, viscous, compressible, incompresible
4
Understand the concept of boundary layers
47
Learning objectives
5
Understand the 3 fundamental physical laws for aerodynamics
Conservations of mass, momentum, and energy
6
Understand why the nozzles of rocket engines, or garden hoses, are shaped
the way they are
Continuity equation: A1 V1 = A2 V2
7
Understand the relation between pressure and velocity along the streamline
V2
V2
Bernoulli’s equation: p1 + ρ 1 = p2 + ρ 2
2
2
Understand the relation between temperature and velocity
V2
V2
Energy equation: cp T1 + 1 = cp T2 + 2
2
2
8
48