drone equations
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宿迁 2015年 3月
航模提高课程
Drone Colloquia
3.27.2015
Drone and Equations
Jun Steed Huang
宿迁 2015年 3月
宿迁 2015年 3月
航模提高课程
Slogans
1. Keep‘em Flying ！ USA fly club
2. Our machine our plans! Wright bros
3. United we achieve! UK fly club
4. ‘One God! One Aim! One Destiny! Africa f ly club.
5. Suqian fly club ：Practical,Innovative,
Cooperative,Dedicative ！
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Fluid = matter that flows under external forces = liquid or gas.
solid liquid gas
inter-mol forces strongest medium weakest
volume fixed fixed variable
shape fixed variable variable
Examples of fluid motion:
• Tornadoes.
• Airflight: plane supported by pressure on wings.
• Gas from giant star being sucked into a black hole.
• Brake fluid in a car’s braking system.
• Breathing: air into lung & blood stream.
Basic Concept
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Fluid DynamicsMoving fluid is described by its flow velocity v( r, t ).
Streamlines = Lines with tangents everywhere parallel to v( r, t ).
Spacing of streamlines is inversely proportional to the flow speed.
Steady flow : ( ) ( ), t =v r v r
Small particles (e.g., dyes) in fluid
move along streamlines.
e.g., calm river.
Example of unsteady flow: blood in arteries ( pumped by heart ).
Fluid dynamics : Newton’s law + diffusing viscosity Navier-Stokes equations
slow fast
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Sport car
Photo shows smoke particles tracing streamlines in a test of a car’s aerodynamic properties.
Is the flow speed greater
(a) over the top, or
(b) at the back?
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Conservation of Mass: The Continuity Equation
Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines.
flow tubes do not cross streamlines.
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Energy Mass Equations
Same fluid element enters & leaves tube:
( )2 22 1
1
2K m v v∆ = −
Work done by pressure upon its entering tube:
1 1 1 1W p A x= ∆
Work done by pressure upon its leaving tube:2 2 2 2W p A x= − ∆
Work done by gravity during the trip: ( )2 1gW m g y y= − −
W-E theorem: 1 2 gW W W K+ + = ∆ ( ) ( )2 21 1 2 2 2 1 2 1
1
2p V p V m g y y m v v− − − = −
1 1p V=
2 2p V= −
Incompressible fluid: 1 2V V V= =m
Vρ =
21
2p v g y constρ ρ+ + =
Bernoulli’s Eq.
Viscosity & other works neglected
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Applications of Fluid Dynamics
Strategy
Identify a flow tube.
Draw a sketch of the situation, showing the flow tube.
Determine two points on your sketch.
Apply the continuity equation and Bernoulli’s equation.
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Steady flow
Conservation of mass:
1 1 1 1m A v tρ= ∆Mass entering tube:
2 2 2 2m A v tρ= ∆Mass leaving tube:
1 1 1 2 2 2A v A vρ ρ=
A v constρ = ρ= ×v AEquation of continuity for steady flow :
Mass flow rate = [ ρ v A ] = kg / s
Volume flow rate = A v const=Liquid : [ v A ] = m3 / s= ×v A
Liquid : flows faster in constricted area.
Gas with v < vs ound: flows faster in constricted area.
Gas with v > vsound : flows slower in constricted area.
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Bernoulli Effect
A ping-pong ball supported by downward-flowing air.
High-velocity flow is inside the narrow part of the funnel.
Bernoulli Effect: p ↓ v ↑
Example: Prairie dog’s hole
Dirt mound forces wind to accelerate over hole
low pressure above hole
natural ventilation
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Flight Lift
Aerodynamic lift
Top view on a curved ball : spin
Blade pushes down on air
Air pushes up (3rd law) Faster flow, lower P : uplift.
Top view on a straight ball : no spin
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Viscosity Turbulence
Smooth flow becomes turbulent.
Viscosity: friction due to momentum transfer between adjacent
fluid layers or between fluid & wall.
B.C.: v = 0 at wall
• drag on moving object
• provide 3rd law force on propellers.
• stabilize flow.
flow with no viscosity
flow with viscosity
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2D Newton Navier-Stokes
0=∂∂+
∂∂+
∂∂
y
F
x
E
t
U
=
tE
v
uU
ρρρ
+−−+−
−+=
xxyxxt
xy
xx
qvuupE
uv
pu
u
E
τττρ
τρρ
)(
2
+−−+−+
−=
yyyxyt
yy
xy
qvuvpE
pv
uv
v
F
τττρ
τρρ
)(
2
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[ ])(. 2250 vueEt +⋅+= ρ
Approximations
∂∂+
∂∂+
∂∂=
y
v
x
u
x
uxx λµτ 2
∂∂+
∂∂+
∂∂=
y
v
x
u
y
vyy λµτ 2
∂∂+
∂∂==
x
v
y
uyxxy λττ
µλ
λµµ
3
23
2
−=
+=′
yTKq
xTKq
y
x
∂∂−=∂∂−=
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Pure Gas
TRp ρ=
TCe v=
2
23
1 CT
TC
+=µ
K
CP pr
µ=
Gas
Sutherland
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Partial Equations
0=∂∂+
∂∂+
∂∂+
∂∂+
∂∂
y
v
x
u
yv
xu
tρρρρρ
021 2
2
2
2
2
=∂∂
∂+−∂∂−
∂∂+−
∂∂+
∂∂+
∂∂+
∂∂
yx
v
y
u
x
u
x
p
y
uv
x
uu
t
u
ρµλ
ρµ
ρµλ
ρ
021 2
2
2
2
2
=∂∂
∂+−∂∂−
∂∂+−
∂∂+
∂∂+
∂∂+
∂∂
yx
u
x
v
y
v
y
p
y
vv
x
vu
t
v
ρµλ
ρµ
ρµλ
ρ
Φ=
∂∂+
∂∂−
∂∂+
∂∂+
∂∂+
∂∂+
∂∂+
∂∂+
∂∂
vrr C
R
y
p
x
p
PyxP
p
y
v
x
up
y
pv
x
pu
t
p2
2
2
2
2
2
2
2
2 ργµρρ
ργµγ
2222
22
∂∂+
∂∂+
∂∂+
∂∂+
∂∂+
∂∂=Φ
y
v
x
u
y
v
y
u
x
v
x
u λµ
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宿迁 2015年 3月
Parabolic Navier-Stokes
In the flow near the object, the dissipation in the tangential direction is much smaller than that in the normal direction. Therefore, the Navier-Stokes equations can be omitted from all the second derivative terms along the main flow direction to obtain the parabolic Navier -Stokes equations.
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Incompressible Newtonian Fluid
Navier - Stokes Equations for Unsteady Flow of Incompressible Newtonian Fluid
For liquid or low-speed moving gases, an incompressible approximation, Dp / Dt = 0, can be used, and the energy equation can be solved separately from the continuous equation and the equation of motion.
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Adiabatic Approximation
Euler Equations for Compressible Ideal Gas Unsteady Adiabatic Flow
Experiments show that in the large Reynolds number flow, the effect of viscosity only in the thin layer near the object is important, and the effect of viscosity in the main flow region excluding the thin layer is negligible, so that the fluid in the main region can be approximated It is considered to be a viscous ideal fluid.
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Compressible ideal fluid unsteady transonic velocityless spin flow
When the object in the ideal fluid for subcritical flight, the entire flow field will be no spin; even when the object for transonic flight, the shock intensity is not large, the entire flow field can also be regarded as a spinless.
( ) ( ) 0222 2222
2
2
=∂∂−+
∂∂+
∂∂−+
∂∂+
∂∂+
∂∂
y
vav
y
uuv
x
uau
t
vv
t
uu
t
ϕ
ux
=∂∂ϕ
vy
=∂∂ϕ
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Small perturbation equations for compressible ideal fluid unsteady transonic velocityless spin flow
If the flow field we study is due to the small perturbations that occur in a homogeneous flow, such as studying a uniform flow around a thin wing, the problem can be further simplified due to the small perturbation assumption.
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Compressible ideal turbulence equation for unsteady subsonic or supersonic spinless flow
( ) 01 2 =∂∂+
∂∂− ∞ y
v
x
uMa
0=∂∂−
∂∂
y
u
x
v
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First order quasilinear partial differential equations
fx
UA
t
U
ii =
∂∂+
∂∂
[ ]TnUUUU ⋅⋅⋅= 21
[ ]Tnffff ⋅⋅⋅= 21
( ) ( )njiAj ,,, ⋅⋅⋅= 21λ iA
( ) 0=− IA iAi λ
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Equation classification
If the n eigenvalues are all complex numbers, the equation is purely elliptic in plane;
If the n eigenvalues are real numbers that are not equal to zero and are not equal to zero, then the equation is purely hyperbolic on the plane;
If the n eigenvalues are all zero, then the equation is parabolic on the plane;
If the n eigenvalue part is a real number, part is a complex number, then the equation is hyperbolic - elliptical, or elliptical in plane;
If the n eigenvalues are all real and part is zero and the part is not zero, then the equation is hyperbolic parabolic or parabolic in plane.
ixt −
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Compressible ideal fluid two - dimensional small subsonic or supersonic spin - free small perturbation equation
( ) 01 2 =∂∂+
∂∂− ∞ y
v
x
uMa
0=∂∂−
∂∂
y
u
x
v
0=∂∂+
∂∂
y
UA
x
U
=v
uU
−
=01
10 βA
21 ∞−= Maβ
0=− IA λ
012 =+ βλ
Subsonic flow, the equation is purely elliptical Supersonic flow, the equation is pure hyperbolic
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宿迁 2015 年 3 月
Euler equation for compressible ideal two - dimensional unsteady adiabatic flow of complete gas
0=∂∂+
∂∂+
∂∂
y
UB
x
UA
t
U
=
p
v
uU
ρ
=
up
u
u
u
A
00
000
100
00
γ
ρρ
=
vp
v
v
v
B
γρ
ρ
00
100
000
00
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Euler equation for compressible ideal two - dimensional unsteady adiabatic flow of complete gas
0=∂
∂+∂∂+
∂∂
t
UD
y
UC
x
U
C Eigenvalueu
vC =21,λ
( )22
222
43au
avuauvC
−−+±
=,λ
Subsonic flow, in the (x, y) plane is hyperbolic Supersonic flow, in the (x, y) plane is hyperbolic elliptical
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Euler equation for compressible ideal two - dimensional unsteady adiabatic flow of complete gas
0=∂
∂+∂∂+
∂∂
t
UD
y
UC
x
U
D Eigenvalue
In the (x, t) plane is pure hyperbolic Similarly, it is purely hyperbolic in the (y, t) plane
uD 1
21 =,λ
auD
±= 1
43,λ
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Navier - Stokes Equations for Two - Dimensional Unsteady Flow of Compressible Viscous Frequent Heat
On any plane containing the time t axis, the Navier-Stokes equation, which is compressible and viscous, which is often non-steady flow of heat completely gas, is a hyperbolic parabolic system and a hyperbolic elliptic equation on the (x, y) plane group.
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The Method of Boundary Value Problem for Elliptic Partial Differential Equation
( ) Dyxy
u
x
u ∈=∂∂+
∂∂
,02
2
2
2
( )yxfu ,| =Γ
( )yxfn
u,=Γ
∂∂
( )yxfhun
uk ,=
+
∂∂
Γ
For oval elliptic partial differential equations, it is required to specify the boundary conditions on the entire boundary of the enclosed area
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The Initial Value and Boundary Value Problem of Hyperbolic Partial Differential Equation
( )0=
∂∂+
∂∂
x
u
t
ρρ
01 =
∂∂+
∂∂+
∂∂
x
p
x
uu
t
u
ρ
02 =∂∂+
∂∂+
∂∂
x
ua
x
pu
t
p ρ
audt
dxC +=+ :
audt
dxC −=− :
udt
dxC =:0
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The Initial Value and Boundary Value Problem of Hyperbolic Partial Differential Equation
x
t
A
A’
B
D
C
C’
C-
0 L
t0
C0
C+
C+
C+
C+
C+
C0
C0
C0
C0
C-
C-
C-
C-
The Initial and Boundary Condition of One - dimensional Unsteady Flow Euler Equation
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The Initial Value and Boundary Value Problem of Parabolic Partial Differential Equation
2
2
x
u
t
u
∂∂=
∂∂ α
( ) ( )xfxu =0,
x
t
0 L
The Initial and Boundary Conditions of One - dimensional Heat Conduction Equation
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The Initial Value and Boundary Value Problem of Parabolic Partial Differential Equation
( )tgu =Γ|
( )tgn
u =Γ∂∂
|
( )tghun
uk =Γ
+
∂∂
Parabolic partial differential equations require that the definite condition be specified on all boundaries of the open region.
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The condition of the boundary given
The appropriate conditions of the boundary conditions and its mathematical processing to stabilize the calculation process necessary conditions;
The specific method of boundary processing may affect the calculation accuracy of physical quantities such as friction, heat flow and so on;
In a detailed simulation of some flow problems, boundary processing will have an effect on the internal structure of the flow field.
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Computational Fluid Dynamics
(1) the actual boundary: they are determined by the nature of the physical problem, which is determined. For example, the solid wall in the outflow problem, the inlet and outlet boundaries in the internal flow problem and the solid walls are the actual boundaries.
(2) artificial boundaries: they are for infinite or semi-infinite areas, or people interested in the scope of the area is much smaller than the artificial introduction. For example, in the calculation of the outflow problem, although the actual area extends to infinity, the outer boundary can only be selected at a distance from the solid boundary.
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The number of physical boundary conditions required for suitability
TypeEuler Navier-Stokes
Supersonic inflow 5 5
Subsonic inflow 4 5
Supersonic outflow 0 4
Subsonic outflow 1 4
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航模提高课程
Drone Colloquia
Thank you for watching this presentation!
谢谢姚成，孙晨旭，邹青，黄佳敏，韩学波，沙龙，周嘉宇的协助。
宿迁 2015年 3月