Bivectors W_{+} and T(W_{+}) each have ½N(N+1) coordinates since W_{+} resides in e_{¥}^{*}
while T resides in e_{0}^{*}.
Thus expressing each T coordinate as a general quadratic form of the W_{+} coordinates requires
(½N(N+1))^{3} scalar coefficients. This is 27 for N=2 and 216 for N=3.
Consider approximating
T(W_{+}) as å_{k=1}^{K}
(W_{+}¿a_{k})Ù(W_{+}¿b_{k})
for 2K 3-vectors a_{1},b_{1}),...,a_{k},b_{k}).
imparted by the fluid to the aircraft is dependant solely on the surface geometry of the
Fluid Mechanics
Introduction
To model the motion of a body through a fluid medium such as air or water we must at least approximate the
effects of the fluid on the body which requires simulation at least in part of the motion of the fluid, which is likely to be complicated and frquently chaotic.
Classical Flows
Classical fluid mechanics typically represents a flow of matter by a Â^{N0} nonunit 1-field
where N_{0} is the number of spacial dimensions with M_{P} representing at a given time t the total momentum per unit
volume passing through a small test volume at spacial position PÎÂ^{N0};
together with scalar 0-field m_{P} representing the density (mass per unit volume) at P (as measured over a small
test volume) , constant for an incompressible fluid. We have M_{p} = m_{p}V_{p} where
V_{p}=m_{p}^{-1}M_{p} is the (nonunit) average spacial velocity vector of the matter at
p= P+te_{4} is an N=N_{0}+1 dimensional spacetime event. V_{p} is undefined when m_{p}=0.
In thermodynamics we also consider the matter at p to have scalar temperature Q_{p}.
The flow satisfies the conservation of matter law (aka. a continuity equation)
if
Ñ_{P}¿M_{p} = -¶m_{P}/¶t = 0
which states that the ammount of matter leaving a small region of volume V at P per unit time
as measured over a small test interval equals the fall
in the ammount of matter m_{P}V at V at P per unit time
We can express the matter conservation law
as
¶ m_{P}/¶ t + Ñ_{P}¿(m_{p}V_{p})
= ¶ m_{P}/¶ t + (V_{p}¿Ñ_{P})m_{p}_{Ñ}
+ m_{p}(Ñ_{P}¿V_{p})
= ((¶ /¶ t) + (V_{p}¿Ñ_{P}))m_{p}
+ m_{p}(Ñ_{P}¿V_{p})
= 0
where (¶ /¶ t) + (V_{p}¿Ñ_{P})
is known as the substantial derivative
corresponding to the rate of change when "following" the flow. Note that this represents the derivative in the direction of the flow
scaled by the speed of the flow.
If we define Â^{N} 1-vectors p=P+te_{4} and v_{p}=V_{p}±e_{4} where N=N_{0}+1 and e_{4} is a (non-relativistic) "temporal" axis perspendicular to Â^{N0} with e_{4}^{2}=±1 then the substantial direction becomes (v_{p}¿Ñ_{p}) with the differentiating scope of the Ñ_{p} extending only rightwards rather than encompassing the v_{p}.
For a steady flow M_{p} = M_{P} with vanishing Ð_{e4} derivatives the matter conservation law becomes (V_{p}¿Ñ_{P})m_{p}_{Ñ} + m_{p}(Ñ_{P}¿V_{p}_{Ñ}) = 0 and for an incompressible flow with m_{p}=m>0 this yields the incompresability condition Ñ_{P}¿V_{p} = 0, ie. the spacial divergence of the N_{0}-D velocity 1-vector is zero. For v_{p}=V_{p}±e_{4} , the incompressability condition is Ñ_{p}¿v_{p}=0.
For an incompressible flow "momentum" and "velocity" fields become effectively equivalent with
M_{p}=mV_{p}.
Classical Pressure
The scalar static pressure s_{p} at p is independant of the macroscopic "net flow" m_{p} at p ,
and stems instead from microscopic "thermal" agitatation of the particles "at" p .
The use of the term "static" here is traditional, and relates to "pressure not due to flow" rather than a restriction on
the variability of s_{p} with p. The pressure s_{p} at p the sum of the static pressure and pressure due to the fluid motion,
Consider a small Â^{3} 3-simplex or 2-sphere V centred at p in a fluid at rest from within which the
fluid has been removed. Random microscopic thermal motion of the fluid will tend to cause molecular impacts
on the surface of the simplex which we can regard as imparting locally normal "inward" momentum uniformly across the boundary surface.
Integrating this inward component across the surface and dividing by the surface area content leads us to
scalar pressure which we can regard as the magnitude of the thermal e_{4}-forces exterted across a small 2-simplex
or 1-sphere centred at p of aribitary 2-blade tangent, divided by the simplex area.
For N_{0}||>3, scalar s_{p} represents the inward thermal forces across a (N_{0}-1)-sphre.
In an anisotropic ideal gas s_{p}V is proportional to nQ_{p} where V=|V| is the volume
of a test 3-simplex V at p containing n molecules of average temperature Q_{p}. n is proportional
to total mass m_{p}V so s_{p} is proportionate to m_{p}Q_{p} and we usually write
s_{p} = Rm_{p}Q_{p} where R is the gas constant for the fluid.
The entropy proportionate to (s_{p}m_{p}^{-g})^{↓}
= s_{p}^{↓} - gm_{p}^{↓}
where dimensionless g is the specific heat ratio
is conserved by fluid elements in the absence of heat conduction so that (v_{p}¿Ñ_{p})s_{p}m_{p}^{-g} = 0
. Entropy has units ( m^{-4})^{↓} so can be measured in m^{↓}
.
Now consider some fluid inside V. The total forces acting on this fluid are
the external macroscopic forces
ò_{V} |d^{3}p| f_{p}
where f_{p}=m_{p}g_{p} + h_{p} is e_{4}-spacial forces exerted
(typically we might have g_{p}=-ge_{3} for uniform vertical gravity)
,
and the static pressure integrated over the boundary
ò_{dV}|d^{2}p|(-n_{p})s_{p}
= -ò_{dV}d^{2}p e_{123}^{-1}s_{p}
= -e_{123}^{-1} ò_{V}d^{3}p Ñ_{P}s_{p}
where n_{p}
= I_{2p}^{-1} e_{123}
= I_{2p} e_{123}^{-1}
is the outward normal at p Î dV .
Classicists typically asssume g_{p}=-Ñ_{p}G_{p} and h_{p}=-Ñ_{p}H_{p} derive from gravitational and
nongravitational scalar 0-potentials.
The total change in momentum incured by the mass m_{p}|V| of fluid as it follows the flow lines
is
ò_{V} d^{3}p (Ñ_{P}¿v_{p})m_{p} so we have
(Ñ_{P}¿v_{p})m_{p} = m_{p}g_{p} + h_{p} - Ñ_{p}s_{p} which for constant density m_{p}=m simplifies
to the momentum conservation law
(v_{p}¿Ñ_{P})v_{p} = g_{p} + m^{-1}h_{p} - m^{-1}(Ñ_{P}s_{P})
aka. Euler's equation when combined with Ñ_{P}¿V_{P}=0 .
For P in Â^{3} we have
(V_{p}¿Ñ_{P})V_{p}_{Ñ} = ½Ñ_{P}(V_{p}^{2}) - V_{p}×(Ñ_{P}×V_{p}_{Ñ})
so we can express the momentum conservation law as
½Ñ_{P}(V_{p}^{2}) - V_{p}×(Ñ_{P}×V_{p}_{Ñ}) + m^{-1}Ñ_{P}s_{p} -
m^{-1}f_{P} = 0 .
[ Proof :
a×(b×c) = (a¿c)b - (a¿b)c Þ
V_{p}×(Ñ_{P}×V_{p}) = (V_{p}¿Ñ_{P})V_{p}_{Ñ} - (V_{p}¿V_{p}_{Ñ})Ñ_{P}
= (V_{p}¿Ñ_{P})V_{p}_{Ñ} - Ñ_{P}(V_{p}_{Ñ}V_{p})
= (V_{p}¿Ñ_{P})V_{p}_{Ñ} - ½Ñ_{P}(V_{p}^{2})
.]
Small peturbation sound waves
Let scalars pressure s_{p} = s_{0}+s_{p} and density m_{p}=m_{0}+m_{p} be small deviations from s_{0} and _wasmup0
Since entropy (s_{p}m_{p}^{-g})^{↓}
= (s_{0}+s_{p})(m_{0}+m_{p})^{-g})
= _presp0m_{0}^{-g} Þ (1+s_{0}^{-1}s_{p})(1+m_{0}^{-1}m_{p})^{-g} = 1
which to first order in s_{0}^{-1}s_{p} and m_{0}^{-1}m_{p}
is s_{p} = s^{2}m_{p} where s=
(m_{0}^{-1}s_{0}g)^{½} .
Now let us suppose that V_{p} is small so that the
matter conservation law
m_{p}^{·} = -(V_{p}¿Ñ_{P})m_{p} - m_{p}(Ñ_{P}¿V_{p}) reduces to
m_{p}^{·} = -m_{0}(Ñ_{P}¿V_{p}) giving
(Ñ_{P}¿V_{p}) » -m_{0}^{-1}m_{p}^{·} .
The
momentum conservation law linearises
to Ñ_{P}s_{p} =
(m_{p}+m_{0})g_{p} + h_{p}
- (m_{p}+m_{0})V_{p}^{·} »
(m_{p}+m_{0})g_{p} + h_{p}
- m_{0}V_{p}^{·}
which gives
Ñ_{P}^{2}s_{p} = (Ñ_{P}¿)((m_{p}+m_{0})g_{p} + h_{p} - (m_{p}+m_{0})V_{p}^{·})
= (Ñ_{P}m_{p})¿g_{p} - m_{0}(Ñ_{P}¿)(V_{p}^{·})
= (Ñ_{P}m_{p})¿g_{p} - m_{0}(Ñ_{P}¿V_{p})^{·}
= (Ñ_{P}m_{p})¿g_{p} + (m_{p}^{·})^{·}
= (Ñ_{P}m_{p})¿g_{p} + (s^{-2}s_{p}^{·})^{·}
Þ
s_{p}^{··} = s^{2}Ñ_{P}^{2}s_{p} so the pressure and desnity perturbations
satisfy the standard 3D wave equation with radial solution
s_{p} = r^{-1}(F(r-st)+G(r+st)) corresponding to an outward wave of amplitude F
of radial speed s and an inward wave of amplitude G and radial speed -s.
s=(m_{0}^{-1}s_{0}g)^{½} thus corresponds to the propagation
speed of density and pressure perturbations from steady state values m_{0} and s_{0} and is known as the speed of sound
in the fluid.
Navier-Stokes
Suppose we combine the static pressure force -s_{p}n_{p} with a viscous force
u((n_{p}¿Ñ_{p})m_{p}_{Ñ} + Ñ_{p}(n_{p}¿m_{p}_{Ñ}))
; reducing to um((n_{p}¿Ñ_{p})v_{p}_{Ñ} + Ñ_{p}(n_{p}¿v_{p}_{Ñ}))
when m_{p}=m.
If we regard n_{p} as varying negligibly with p compared to the variation of v_{p}
(eg. when over a flat surface) then we can view this as the n_{p} directed flow derivative plus the gradient of the n_{p}-directed flow-speed, equally wieghted
by scalar u, leading to the Navier-Stokes equation
(v_{p}¿Ñ_{p})m_{p}
= -m_{p}(Ñ_{p}G_{p}) - (Ñ_{p}s_{p}) - (Ñ_{p}H_{p}) + (v_{p}¿Ñ_{p})m_{p}_{Ñ}v_{p}
+ uÑ_{p}^{2}m_{p} .
For incompressible flow m_{p}=m this becomes
(v_{p}¿Ñ_{p})v_{p}
= -(Ñ_{p}G_{p}) - m^{-1}((Ñ_{p}s_{p}) + (Ñ_{p}H_{p})) + uÑ_{p}^{2}v_{p}
where scalar u is the kinematic viscosity. Water has kinematic viscosity 10^{-6} m^{2}s^{-1}
at 15 ^{o}C while air is fifteen times that, olive oil a hundred. Treacle has viscosity of roughly 1.2×10^{7}
m^{2}s^{-1} at 15 ^{o}C, falling rapidly with temeprature.
Vorticity
For an incompressible flow under gravity, Navier-Stokes becomes
(Ñ_{p}¿v_{p})v_{p} = -Ñ_{p}G_{p} - m^{-1}((Ñ_{p}s_{p})
+ uÑ_{p}^{2}v_{p}
and since (v_{p}¿Ñ_{p})v_{p} = v_{p}¿(Ñ_{p}Ùv_{p}) + ½Ñ_{p}(v_{p}^{2})
we have v_{p}¿w_{p} = -Ñ_{p}E_{p} + uÑ_{p}^{2}v_{p} where
E_{p} = ½v_{p}^{2} + G_{p} + m^{-1}s_{p}
and w_{p} º Ñ_{p}Ùv_{p} has Ñ_{p}¿w_{p}=0.
Hence
Ñ_{p}Ù(v_{p}¿w_{p}) = uÑ_{p}Ù(Ñ_{p}^{2}v_{p})
= uÑ_{p}^{3}.v_{p} .
But Ñ_{p}Ù(v_{p}¿w_{p}) =
(Ñ_{p}¿(v_{p}Ùw_{p}))i where
(N-2)-vector kinematic vorticity w_{p} º w_{p}i^{-1} = (Ñ_{p}Ùv_{p})^{*} so we have Ñ_{p}¿(v_{p}Ùw_{p}) = 0
[ Proof :
Ñ_{p}Ù(v_{p}.w_{p}) = (Ñ_{p}¿(v_{p}Ù(w_{p}i^{-1})))i
= (Ñ_{p}¿(v_{p}Ùw_{p}))i
.]
For incompressible flow,
Ñ_{p}¿(v_{p}Ùw_{p})
= (v_{p}¿Ñ_{p})w_{p}_{Ñ}
- v_{p}_{Ñ}Ù(w_{p}.Ñ_{p})v_{p}_{Ñ}
so we have
geometric vorticity equation
(v_{p}¿Ñ_{p})w_{p}
= (w_{p}.Ñ_{p})Ùv_{p} + u Ñ_{p}¿(Ñ_{p}^{2}v_{p}i^{-1})
with the differentiating scope taken rightwards only.
For N=3 inviscid flow (u=0) this reduces to
(v_{p}.Ñ_{p})w_{p} = (w_{p}.Ñ_{p})v_{p} with 1-vector vorticity w_{p} = (Ñ_{P}ÙV_{P})^{*} = Ñ_{P}×V_{P} and is known as the third Helmholtz vortex theorem.
[ Proof : (v_{p}¿Ñ_{p})w_{p}_{Ñ}
+ (Ñ¿v_{p}_{Ñ})w_{p}
- v_{p}Ù(Ñ_{p}¿w_{p}_{Ñ})
- v_{p}_{Ñ}Ù(Ñ_{p}¿w_{p})
reduces to result since Ñ_{p}¿w_{p} = Ñ_{p}Ùw_{p} = Ñ_{p}ÙÑ_{p}Ùv_{p} vanishes as does
Ñ_{p}¿v_{p} .
Also u(Ñ_{p}Ù(Ñ_{p}^{2}v_{p}))i^{-1}
= u(Ñ_{p}¿((Ñ_{p}^{2}v_{p})i^{-1})
.]
Thus for N_{0}=3 the flow-directed derivative of the vorticity (v_{p}¿Ñ_{p})w_{p}_{Ñ} equals the vorticity-directed derivative of the flow (w_{p}¿Ñ_{p})v_{p}_{Ñ} ; while for N_{0}=2 steady flows we have scalar vorticity preserved along streamlines (Ñ_{P}¿V_{P})w_{p}=0 since (w_{p}¿Ñ_{p})v_{p}=wÐ_{e3}v_{p}=0.
When f_{p}=0, integrating along a streamline gives
Bernoulli's equation s_{p} + ½mv_{p}^{2} = s_{0} ,
where scalar s_{0} is known as the stagnation pressure.
Bernoulli's equation applies when m_{p} is constant and the flow is static. More generally
E_{p} = s_{p} + mG_{p} + H_{p} + ½mv_{p}^{2} is constant along the streamline if m_{p}=m and w_{p}=0 along it.
[ Proof :
s_{p1} = s_{p0} + ò_{t0}^{t1} dt v_{p}¿(Ñ_{p}s_{p})
= s_{p0} + ò_{t0}^{t1} dt v_{p}¿
(-m_{p}(Ñ_{p}G_{p}) - (Ñ_{p}H_{p}) - (Ñ_{p}¿v_{p})m_{p}_{Ñ}v_{p} -
m_{p}(Ñ_{p}¿v_{p})v_{p} )
= s_{p0} + m(G_{0}-G_{1})
(H_{0}-H_{1})
- m ò_{t0}^{t1} dt v_{p}¿((Ñ_{p}¿v_{p})v_{p} )
= s_{p0} + m(G_{0}-G_{1})
(H_{0}-H_{1})
- m ò_{t0}^{t1} dt v_{p}¿(v_{p}¿w_{p} + Ñ_{p}v_{p}^{2} )
= s_{p0} + m(G_{0}-G_{1})
(H_{0}-H_{1})
+ ½m (v_{0}^{2} - v_{1}^{2}) .
Hence s_{p} + mG_{p} + H_{p} + ½mv_{p}^{2} is constant along the streamline.
.]
Circulation
The geometric M-circulation about a closed M-curve C_{M} of a flow m_{p}=m_{p}v_{p}
is the <M-1;M+1>-vector G_{C1} º ò_{CM} d^{M}p v_{p} .
Circulation is thus usually considered to be an integration of velocity rather than momentum and so has units
m^{M+1} s^{-1}.
The 1-circulation about a closed 1-curve (ie. a loop) C_{1}
is thus the <0;2>-vector G_{C1} º ò_{C1} dp v_{p}
and the term circulation traditionally refers to the scalar part
G_{C1} º ò_{C1} dp¿v_{p} .
For N_{0}=2, C_{1} bounds a "solid" flat surface region C_{2} and for an irrotational flow with
v_{p}=Ñ_{p}f_{p}
defined over C_{2} we have
G_{C1} º ò_{dC2} dp v_{p}
= ò_{dC2} dp Ñ_{p}f_{p}
= ò_{d2C2} d^{0}p f_{p} = 0 .
Hence for N_{0}=2 the scalar circulation vanishes provided the region enclosed
by C_{1} contains no regions absent of flow or at which Ñ_{p}Ùv_{p} is nonzero. In particular the scalar circulation around a
solid lamina need not vanish, but is independant of enclosing path C_{1}. Similarly a vortex point with nonzero Ñ_{p}Ùv_{p} within C_{2} will contribute a fixed
"residue" circulation independant of the enclosing path.
|&| ò_{C2}f_{p}|d^{2}p| .
The geometric generalisation of Stoke's law
ò_{dCN0-1} d^{N0-2}p.¦(p)
= (-1)^{N} ò_{CN0-1} (ÑÙ¦(p_{Ñ})).d^{N0-1}p
means that the (N-3)-vector component of the (N_{0}-2)-circulation around the (N_{0}-2)-curve boundary of an
open hypercurve C_{N-1} over which
v_{p} is defined equals the vorticity integration over C_{N0-1}
(-1)^{N0} ò_{CN0-1} (ÑÙv_{p}).d^{N0-1}p .
Thus if v_{p} is defined and irrotational over C_{N0-1} the (N_{0}-3) component of G_{dCN0-1}
vanishes.
Thus for N_{0}=3 we have
G_{dC2} º ò_{C1} dp¿v_{p}
= -ò_{C2} (ÑÙv_{p}).d^{2}p
and so if G_{C1} is nonzero for a closed 3D 1-curve (ie. a path loop) and C_{2} is any
3D hypercurve spanning (bounded by) C_{1} over which v_{p} is everywhere defined, then v_{p} must be irrotational over one or more
regions of C_{2}.
If C_{N0-1}=dC_{N0} is the hypercurve boundary of an open set C_{N0}
over which v_{p} is defined
then the <N_{0}-2;N_{0}>-vector hypercirculation G_{CN-1}
º ò_{CN-1} d^{N-1}p v_{p}
= ò_{CN} d^{N}p Ñ_{p}v_{p}
vanishes for a static incompressible irrotatuonal flow with Ñ_{p}v_{p}=0.
If C_{N} containis regions over which v_{p} in undefined or zero then
G_{CN-1} may be nonzero but tends to be indendant of C_{N-1}
in that we can deform C_{N-1} without altering G_{CN-1} provided we do not move
C_{N-1} across any undefined v_{p} regions .
The hypercirculation about a (N_{0}-1)-curve C_{N0-1} is the <N_{0}-2;N_{0}>-vector
G_{CN-1} º ò_{CN-1} d^{N-1}p v_{p}
=
ò_{CN0-1} ((-1)^{N0-2} ¯_{dN0-1p}(v_{p}) _dNm1 + (-1)^{N-1}^_{dN0-1p}(v_{p}) d^{N0-1}p)
= (-1)^{N0}ò_{CN0-1} (¯_{dN0-1p}(v_{p}) - ^_{_dN0m1}(v_{p})) d^{N-1}p
= (-1)^{N0}
(ò_{CN0-1} ¯_{dN0-1p}(v_{p})¿d^{N0-1}p)
- ò_{CN0-1} ^_{dN0-1p}(v_{p})Ùd^{N0-1}p)
.
When v_{p}=Ñ_{p}f_{p} the (N_{0}-2) component
ò_{CN0-1} ¯_{dN0-1p}(v_{p})¿d^{N-1}p
= ò_{CN0-1} ¯_{dN0-1p}(Ñ_{p}f_{p})¿d^{N0-1}p
= ò_{CN0-1} (¯_{dN-1p}(Ñ_{p})f_{p})¿d^{N0-1}p
º ò_{CN0-1} (Ñ_{p}_{[CN-1]}f_{p})¿d^{N0-1}p
= ò_{dCN0-1} f_{p} d^{N0-2}p
vanishes for closed C_{N0-1} (or has a constant (N_{0}-2)-vector value comprising residues at enclosed poles).
The pseudoscalar hypercicrulation component
ò_{CN0-1} ^_{dN0-1p}(v_{p})Ù_dN0m1
= i ò_{CN0-1} (v_{p}¿n_{p})|_dN0m1|
measures the outflow across C_{N0-1} and vanishes for a steady incompressable flow.
Otherwise for small |C_{N0-1}| it approximates
i|C_{N0-1}|(Ñ_{P}¿V_{p}) =
i|C_{N0-1}|Ñ_{P}^{2}f_{p} evaluated at any particular p enclosed by C_{N0-1}.
Reynolds Number
It is frequently the case in Navier-Stokes dynamics that one of (v_{p}¿Ñ_{p})v_{p} and uÑ_{p}^{2}v_{p} will dominate to
the extent that the other may be neglected and far simpler equations solved. The Reynolds number
u^{-1}ua where u is a typical flow speed and a a characteristic length for the problem
provides a rough indicator of the likely ratio of the magntiudes of the inertial (v_{p}¿Ñ_{p})v_{p} term to the viscous uÑ_{p}^{2}v_{p} term;
a high Reynold's number favouring the inviscid approximation forcing u=0.
This can fail when the large second derivatives present in thin boundary layers "seperate" from the boundary and signifcantly effect the flow far from the boundary.
Complex Potential
The complex potential is used to emulate incompressable, irrotational, usually steady, Â^{2} planar flows.
Stream Function
Setting scalar stream function y_{p} º
ò_{C} dV^{2} V^{1}
- ò_{C} dV^{1} V^{2} for any Â^{2} path C from 0 to p
yields
V^{1} = Ð_{e2}y_{p} ; V^{2} = - Ð_{e1}y_{p} ; ie.
V_{p} = e_{12}(Ñ_{P}y_{p})
= (Ñ_{P}y_{p})e_{12}^{-1}
= Ñ_{p}×(y_{p}e_{3})
= (Ñ_{P}Ù(y_{p}e_{3}))e_{123}^{-1}
with V_{p}^{2} = (Ñ_{p}y_{p})^{2} .
If y_{p} is any analytic real scalar field over Â^{2}
satisfying this then its analyticity provides incompressability condition Ð_{e1}V^{1} + Ð_{e2}V^{2} = 0, ie.
Ñ_{P}V_{p}=0 and also Ñ_{p}^{2}y_{p} = 0.
Hence V_{p} = (Ñ_{P}(y_{p}e_{3}))e_{123}^{-1} = -Ñ_{P}y_{p}e_{12} .
V_{p}^{2} = (Ð_{e2}y_{p})^{2} + (-Ð_{e1}y_{p})^{2} = (Ñ_{p}f_{p})^{2}
Also note that Ð_{Vp} y_{p} = (V_{p}¿Ñ_{p})y_{p} = 0 so y_{p} is constant when following the flow.
Such 1-curves along which y_{p} is constant are known as streamlines.
Velocity Potential
For a steady irrotational incompressible flow with Ñ_{p}ÙV_{p} = 0 we can constuct a 0-potential
f_{p} = ò_{C} dp¿V_{p} for any path C from 0 to p so that V_{p}=Ñ_{p}f_{p} , ie.
V^{i} = e_{i}^{2}Ð_{ei}f_{p} . If the flow is conserved we also have Ñ_{p}¿V_{p}=0 so Ñ_{p}V_{p} =
Ñ_{p}^{2}f_{p} = 0.
Setting F_{p} º f_{p}(Ñ_{P}f_{p}) we have
Ñ_{P}F_{p} = (Ñ_{P}f_{p})^{2} + f_{p}Ñ_{P}^{2}f_{p} = (Ñ_{P}f_{p})^{2}
= V_{p}^{2} hence for constant density flow the kinetic energy within a volume
½mò_{V} d^{3}p V_{P}^{2} =
½mò_{dV} d^{2}p f_{p}(Ñ_{P}f_{p})
with pseudoscalar part giving divergence theorem
½mò_{V} |d^{3}p|V_{P}^{2}
= -½m ò_{dV}|d^{2}p|f_{p}Ð_{n}f_{p} where n is the outward normal to the (N_{0}-1)-curve
integration surface dV .
Thus in an incompressibe incompressable flow, we can deduce the kinetic energy insode a volume of fluid
from f_{p}Ð_{n}f_{p} over the blundary of V
Complex representations of 2D Flows
When N_{0}=2 then by taking i=e_{21}=-e_{12} and setting z = pe_{1} = x^{1}+x^{2}e_{21} = x^{1}+x^{2}i ; v(z) = V_{p}e_{1} = V^{1}+V^{2}e_{21} ;
we can work in commtuing algebra C @ Â_{2 +}.
We define automorphic conjugation x^{^} º e_{1}xe_{1} negating e_{2} and e_{12} but preserving 1 and e_{1} so
corresponding to complex conjugation over Â_{2 +} and reflection in x^{2}=0 over Â^{2}, giving innerproduct
a.b = Real(a^{^}b) and a^{2} = a^{^}a =
|a|_{+}^{2} .
Complex Potential
The Complex 0-potential f_{p} º f_{p} + iy_{p}
satisfies Ñ_{P}^{2}f_{p} = 0 since
Ñ_{P}^{2}f_{p} and Ñ_{P}^{2}y_{p} both vanish.
It is defined for x=P where P Î Â^{N0}=Â^{2}
but by defining z=x^{1}+ix^{2} = pe_{1} where p=x^{1}e_{1}+x^{2}e_{2}
and taking i=-e_{12} we can regard
f_{p} = f_{z,t} as a t-dependant regular function mapping C®C
whose conjugated derivative provides flow v(z) = V_{p}e_{1} = f'(z)^{^} .
By virtue of its construction, the complex potential satisfies the
Cauchy-Riemann equations
and so any regular complex function provides a complex potential
with a direction-independant derivative f'(z,t) º (d/dz)f(z,t) =
df/dx^{1} + idy/dx^{1} =
V^{1} - iV^{2} = v^{^}(z,t) .
Hence V_{p} = f'^{^} = f^{^}' where ' denotes regular differention with resepect to z
and so V_{p}^{2} = |f'(z)|_{+}^{2} .
A regular complex 0-potential f_{p}(t) with Ñ_{p}^{2}f_{p}=0 thus fully embodies an incompressible irrotational 2D flow
with spacial derivative f' equal to the complex conjugate of the flow, and streamlines indicated by
constant Imag(f). If f_{p} is independant of t then the flow is steady.
Apart from at z=0, the reciprocal potential f_{p}(t)^{-1} is also regular
with (f_{p}(t)^{-1})' = -f_{p}(t)^{-2} f'_{p}(t) . The
f_{p}(t)^{-2} factor is complex and both redirects and rescales the derivative.
Similarly f(lz^{a}) has flow
(f'(lz^{a})la z^{a-1})^{^} which is
(la z^{a-1})^{^} times the flow due to f(z).
For a=-1 we have f(lz^{-1}) giving flow
-(lz^{-2})^{^} times the flow due to f(z).
Example: Uniform Flow
The uniform flow of speed A and direction subtending a with the horizontal is given by
v(z,t) = a = A(ia)^{↑}
and has f_{p} = a^{^} z = A(-ai)^{↑} z
where p=(x^{1},x^{2})=[r,q] and z=x^{1}+ix^{2}.
Example: Line Vortex
Consider the circular flow u_{p} = gr^{-1}e_{q} , ie.
v(z,t) = g|z|_{+}^{-2}iz
= gi(z^{-1})^{^}
= (-giz^{-1})^{^}
having circulation
2pg
|+ e_{12} ò_{0}^{a} dz ò_{0}^{2p} dq gq z
= 2pg(1+ e_{12}½pa^{2}).
.
y_{p}=-gr^{↓} while the velocity potential is f_{p}=gq yielding
f_{p}
= -gi(z)^{↓}
Setting G=2pg we have line vortex potential
f_{p} = G(2pi)^{-1}(z)^{↓} with circulation
G(1+e_{12}½pa^{2}) about any loop enclosing r=a
and associated flow u_{p} = (2p)^{-1}Gr^{-1}e_{q} tending to zero at r=¥ .
.
The r^{-1} factor on the speed means that the angular momentum m_{p}pÙV_{p} = m_{p}ge_{r}e_{q}
= m_{p}(2p)^{-1}Ge_{12}
= m_{p}(2pi)^{-1}G
about 0 at p is independant of p for incompressible flow m_{p}=m.
Note that f(lz^{-1}) = G(2pi)^{-1}(l^{↓} - z)^{↓}
is the potential for circulation -G plus an irrelevant constant.
For large D, f_{p} = G(2pi)^{-1}(z-D)^{↓} approximates uniform flow
v(0)^{^} = (2p)^{-1}GD^{-1}i(-D)^{~}
= (2pi)^{-1}GD^{-1} so
f_{p} = a^{^}D(z-D)^{↓} provides flow close to
a for r<<D.
Uniform Â^{2} Flow Past a Circular Boundary
If f(z) is the complex potential of a flow having no singularities for |z|<a
then
f(z) + (f(a^{2}z^{^}^{-1}))^{^}
= f(z) + (f(a^{2}|z|_{+}^{-2}z))^{^}
has the same singularities as f(z) over |z|>a
and is purely real when |z|=a (ie. y_{p}=0 when r=a), so the circle r=a
is a streamline.
From this we deduce that
f(z) = a^{^}z + (a^{^}a^{2}z^{^}^{-1})^{^}
= a^{^}z + aa^{2}z^{-1}
has flow f'(z)^{^} approaching
a for large r but with zero e_{r} component when r=a .
By adding a line vortex potential
f(z,t) =
a^{^}(z-c) + a^{2}a(z-c)^{-1}
+ G(2pi)^{-1} (z-c)^{↓}
= A((-ai)^{↑}(z-c) + a^{2}(+ai)^{↑}(z-c)^{-1}) + al_{a}i^{-1}(z-c)^{↓})
where
a=A(ai)^{↑} and
dimensionless
scalar
l º l_{r} º l(r,G,A) º (2pr)^{-1} A^{-1} G
= r^{-1}al_{a}
we obtain the complex potential for a flow
v(z) = f'(z)^{^}
= A((ai)^{↑} - a^{2}(z-c)^{-2}^{^}(-ai)^{↑})
- G(2pi)^{-1}(z-c)^{-1}^{^}
= A((ai)^{↑} - a^{2}(z-c)^{-2}^{^}(-ai)^{↑}
+ al_{a}i(z-c)^{-1}^{^} )
that approaches uniform flow a=A(ai)^{↑} as r®¥
but has circle r=a as a y=G(2p)^{-1}a^{↓} streamline.
Taking a=0, c=0 for a uniform horizontal flow Ae_{1} at infinity
around the origin centered circle we have
f(z) = A(z+a^{2}z^{-1}) + G(2pi)^{-1} z^{↓}
= A(z+a^{2}z^{-1} + al_{a} z^{↓} i^{-1})
= A(r(iq)^{↑} + r^{-1}(-iq)^{↑}a^{2}) ) + G(2pi)^{-1}(r^{↓}+qi)
= Ar(1+(r^{-1}a)^{2}) cos(q) + (2p)^{-1}qG
+ Ar(1-(r^{-1}a)^{2}) sin(q)i - (2p)^{-1}r^{↓}Gi
so our a=0 scalar potential and stream functions are
f(r,q) = Ar(1+(r^{-1}a)^{2}) cos(q) + (2p)^{-1}qG ; y(r,q) = Ar(1-(r^{-1}a)^{2}) sin(q) - (2p)^{-1}r^{↓}G . V_{p} = A( (l - (1+(r^{-1}a)^{2}) sinq)e_{q} + (1-(r^{-1}a)^{2}) cosqe_{r} ) . V_{p}^{2} = A^{2} ( l(r)^{2} - 2l(r)(1+(r^{-1}a)^{2}) sin(q) + 1+(r^{-1}a)^{4}-2(r^{-1}a)^{2} cos(2q) ) . f(lz^{-1}) = A((lz^{-1})+a^{2}(l^{-1}z) - al_{a} z^{↓} i^{-1}) + al_{a} l^{↓} i^{-1}) provides a flow of strength Aa^{2}l about a circular boundary of radius la^{-1} with circulation -G. Along the boundary r=a we have constant y = -(2p)^{-1}a^{↓}G and V_{p} = A(l(a) - 2 sinq)e_{q} parallel to the boundary as required and vanishing at two stagnation points q= sin^{-1}(½l(a)) providing l(a)<2 (ie. G < 4paA) . For l(a)>2 we have just one stagnation point at q=±½p provided l(r) = ±(1+(r^{-1}a)^{2}) Û (r^{-1}a)^{2} ± l(a)(r^{-1}a) + 1 = 0 Û (r^{-1}a) = ½(-/+ l(a) ± (l(a)^{2}-4)^{½}) Û (r^{-1}a) = ½l(a) + ((½l(a)^{2}-1)^{½} . |
Analytically we will consider the generalised circular flow
f(z) = ab(z-c) + a^{2}a^{^}b^{-1}(z-c)^{-1} + G(z-c)^{↓} + Gb
with
f'(z) =
ab - a^{^}a^{2}b^{-1}(z-c)^{-2} + G(z-c)^{-1}
and
f"(z) =
2a^{^}a^{2}b^{-1}(z-c)^{-3} - G(z-c)^{-2}
but in practical cases we require purely imaginary G since the multivalued imaginary component
of z^{↓} = r^{↓} + qi makes for disconnected streamlines if left within y.
Further, we only have a y(z) = Imag(G)(r^{↓}) circular streamline if
b(z-c) = (a^{2}b^{-1}(z-c)^{-1})^{^} which requires
a^{2} = |b|_{+}^{2}|z-c|_{+}^{2}
so we invariably take take a^{2}=a^{2} positive real.
The total force exerted on the r=a boundary by the fluid is -mAGe_{2},
or more generally
mAGae_{12}^{-1} where
a=ae_{1}=A(ae_{21})^{↑}e_{1} is the background flow.
[ Proof : Around the r=a streamline we have
V_{p}^{2} = A^{2}(l_{a} - 2 sin(q))^{2}
and in the absence of gravity (G_{p}=0) and other forces (H_{p}=0) we have
s_{0} = s_{p} + ½mv_{p}^{2}
Þ m^{-1}s_{p} = m^{-1}s_{0} - ½V_{p}^{2}
= m^{-1}s_{0} -
A^{2}(½l_{a}^{2} + 2l_{a} sin(q) - 2 sin(q)^{2}) .
The force exerted on a small element of the boundary is -s_{p}a dqe_{r} having e_{2} component
-s_{p} sin(q)a dqe_{2} and integrating this around the boundary gives
zero total e_{1} force component and a total e_{2} force of
-m ò_{0}^{2p} dq a sin(q)
A^{2}(½l_{a}^{2} + 2l_{a} sin(q) - 2 sin(q)^{2})
= -maA^{2}2l_{a} ò_{0}^{2p} dq
sin(q)^{2})
= -2pamA^{2}l_{a}
= -mAG
.]
Thus there is no "drag" on a circular boundary in a steady uniform inviscid incompressible irrotational flow.
In reality there is drag, of course. Partly because actual flows are viscid but mostly because they are
unsteady and rotational.
Circular Â^{2} Flow Past a Circular Boundary
Taking f(z) + f(a^{2}z^{^}^{-1})^{^}
= f(z) + f(a^{2}|z|_{+}^{-2}z)^{^}
= f(z) + f(a^{2}|z^{-1})^{^}
for f_{p} = a^{^}D(z-D)^{↓} provides
f_{p} = a^{^}D(z-D)^{↓}
+ (a^{^}D(a^{2}z^{-1}-D)^{↓})^{^}
= a^{^}D(z-D)^{↓}
+ aD^{^}((a^{2}z^{-1}-D)^{^})^{↓}
f(z) = b(z-a)_logncj + G(z-c)^{↓}
has v(0)
= -(ba^{-1} + Gc^{-1})
= a^{^} provided
b = a(Gc^{-1} - a^{^}) .
Taking a muich larger than c so that the
a(Gc^{-1} - a^{^})(z-a)^{↓} term
approximates constant flow a^{^} near 0 and c .
Â^{N} flow around hypershpehrical boundary
An analagous result for N>2 is that
y(p) = ½(f(p-c) + (f(a^{N-2}|p-c|^{2-N}(p-c))^{»})
has Ñy(p) = ½(Ñf)(p-c)
+ ½a^{N-2}|p-c|^{2-N}(Ñf)(a^{N-2}|p-c|^{2-N}(p-c))^{»}
since Ñ|x-c|^{2-N} vanishes.
Over |p-c|=a we have
y(p) = ½(f(p-c)+(f(p-c))^{»})
and Ñy(p) = ½((Ñf)(p-c) + ((Ñf)(p-c))^{»}) .
Biimpulse exerted on a 2D boundary
The D'Alembert paradoxical result of zero drag in an inviscid incompressible steady flow around a circular boundary extends to general 2D boundaries
and provides that the force exerted on a 2D body by uniform (at infinity) flow v is
rGe_{12}v perpendicular to v.
Thus for steady inviscid irrotational flow the shape of the 2D boundary is irrelevant to the direction of
the force, but determines its magnitude by fixing the scalar circulation G.
Force perpendicular to fluid velocity is known as lift reagrdless of whether it is "up" or "down"
with regard to e_{3}_{Q} or gravity, eg. a diving plane "lifts" horizontally,
and F=rGe_{12}v equivalent to F=-rGiv is known as the Kutta-Joukowski Lift Theorem.
Since the force derives from the pressure which is determined by V_{p}^{2}, the lift remains the
same if we negate the flow, with both v and G changing sign.
Suppse we have a 2D 1-loop C=p(t) where tÎ[0,L] of total length L that is the y=y_{0} streamline of a steasdy flow.
At a point p(t) = ze_{1} on the boundary C we have a velocity V_{p}=ve_{1}=f'(z)^{^}
and force element dF =
-s_{p}e_{21}dp
= (½r_{p}V_{p}^{2} - s_{0})e_{21}dp
= ½r_{p}V_{p}^{2}e_{21}dtV_{p}^{~} - s_{0}e_{21}dp
= ½r_{p}v^{^}ve_{21}dtv^{~}e_{1} - s_{0}e_{21}dze_{1}
= -½r_{p}v^{2}v^{^}^{~}e_{2}dt - s_{0}e_{2}dz^{^}
= -½r_{p}v^{2}dz^{^}e_{2} - s_{0}e_{2}dz^{^}
Complex force dF =
dFe_{1} =
-½r_{p}v^{2}dz^{^}i - s_{0}idz^{^} with conjuate
dF^{^} =
½ir_{p}(v^{^}^{2})dz + s_{0}dz
= ½ir(z)f'(z)^{2}dz + s_{0}dz
so the net conjugate force exerted on the boundary is
ò_{C} dF ^{^} =
½irò_{C} dz f'(z)^{2} , which is known as Blasius's Theorem.
If f'(z)^{2} is expressed as a Laurent series
f'(z) = å_{k=-n}^{¥} a_{k}(z-c)^{k}
for n³1 about some c then traditional complex residue calculus
provides ò_{C} dz f'(z)^{2} = 2pia_{-1} ,
though of course if there are multiple poles inside C we have multiole contributory residues.
The moment of dF about c is (p-c)ÙdF = ((p-c)dF)_{<2>} -
= ((z-c)e_{1}dFe_{1})_{<2>}
= ((z-c)(dF)^{^})_{<2>} so the total moment about c is
(½irò_{C} dz (z-c)f'(z)^{2})_{<2>}
= ½irReal(ò_{C} dz (z-c) f'(z)^{2} )
= ½irReal(2pia_{-2})
= -pirImag(a_{-2})
in the event of a single pole at c.
Unlike the lift, the torque does not in general vanish when G=0.
The deflector or stress tensor
v_{p}u_{p}^{-1} deflecting bivelocity u_{p} to bivelocity
v_{p} is naturaly parameterised over complex time as
(t(v_{p}u_{p}^{-1})^{↓})^{↑} u_{p} .
Letting
u_{p}(t) = (t(t)(v_{p}u_{p}^{-1})^{↓})^{↑} u_{p}
where t(t) is a path from t(T) to t(T+d).
The aerodynamic profile of a body at a particular mach speed S
can be characterised by directonal deflection r_{p}_vdp_udin
Conformal flow warping
Given a conformal map Z = f(z) and a complex potential f(z) we have complex potential
F(Z) = f(f^{-1}(Z)) defining a flow in the "warped" Z space
V(Z) = F'(Z)^{^} º dF/dZ^{^}
= (df/dz dz/dZ )^{^}
= (df/dz (dZ/dz)^{-1})^{^}
= v(z) (¦'(z)^{-1})^{^}
= v(z) (¦^{-1}'(Z))^{^}
.
The Joukowski transformation Z = ¦(z) = (z-d)+f^{2}(z-d)^{-1}
with dZ/dz = 1 - f^{2}(z-d)^{-2} ;
(d/dz)^{2}Z = 2f^{2}(z-d)^{-3};
and
bivalued inverse
z = ¦^{-1}(Z) = ½(Z ± (Z^{2}-4f^{2})^{½}) + d .
We frequently take real f^{2}=f^{2} real and d=0.
¦ maps z=d±f to Z=±2f and circles to a general "aerofoil like"
family of 2D 1-curves including ellipses and ridged teardrop shapes.
To make ¦^{-1}(Z) single valued, we choose the square root q=(Z^{2}-4f^{2})^{½}
"nearest" to Z in that q^{^}_{*}Z ³ 0 and refer to
¦^{-1}_{+}(Z) = ½(Z+q) + d acting like a d displacement for large r
as the principle Joukowski inverse ; and to
¦^{-1}_{-}(Z)=½(Z-q) + d = f^{2}(¦^{-1}_{+}(Z)-d)^{-1} + d
acting like f^{2}Z^{-1} + d
= f^{2}|Z|_{+}^{-2} Z^{^} + d
for large Z as the secondary Joukowski inverse.
There are discontinuities in ¦^{-1}_{+} and ¦^{-1}_{-} along the line segment connecting
Z = ± 2f acrioss which ¦^{-1}_{+} and ¦^{-1}_{-} "switch values".
A Joukowski aerofoil is the image under ¦ of a circle |z-c|_{+}=a, usually one containing
d-f inside it and c+f either inside the circle or on it.
Havning d+f on the circle requires c = d+f + a(ig)^{↑}
and |d-f-c|_{+} £ a
whence cos(p+g-z) ³ a^{-1}f where f=f(iz)^{↑}.
A Joukowski aerofoil has boundary
|¦^{-1}(Z)-c|_{+} = a but there is a difficulty knowing
which inverse to use. If c lies on the line connecting d±f then the aerofoil
is symmetric and convex and is all "carried" back from Z space
to z space by ¦^{-1}_{+}. But if c lies sufficiently far from the
the d±f connecting centre line, part of the boundary is concave and the region between this and the centre line
is carried by ¦^{-1}_{-} .
The actual condition for the interior of the aerofoil is
|¦^{-1}_{+}(Z)-c|_{+} £ a
&
|¦^{-1}_{-}(Z)-c|_{+} £ a
and we can move the discontinuity from the centre line portion exterior to the aerofoil to the boundary of the aerofoil
by switching to ¦^{-1}_{+} when |¦^{-1}_{-}(Z)-c|_{+} £ a .
¦^{-1}_{±}'(Z) = ½(1 ± q^{-1}Z)
and ¦^{-1}_{±}"(Z) = ±½q^{-1}(1 - q^{-2}Z^{2})
so ¦^{-1}_{+}'(Z)+¦^{-1}_{-}'(Z)=1 and
¦^{-1}_{+}"(Z)+¦^{-1}_{-}"(Z)=0 .
Â^{2} Flow Past a sharp edged 2D Boundary
Joukowski-Kutta Condition
The Joukowski warped flow velocity V(Z) = v(z) (1-f^{2}(z-d)^{-2})^{-1}^{^}
is infinite only when v(z) is or when (z-d)^{2} = f^{2} with z lieing outide the boundary
and v(z) nonzero. Thus if one of z=d±f lies outside
a given closed y(z) streamline conisdered to be a boundary,
then
only by choosing the circulation G to ensure a stagnation
point f'(z)=0
at that z can we eliminate an aphysicsl point of infinite flow.
For a generalised circular flow
f(z) = a(z-c) + a^{2}a^{^}(z-c)^{-1} + G(z-c)^{↓}
with f'(z) = a - a^{2}a^{^}(z-c)^{-2} + G(z-c)^{-1}
finite everywhere but at z=c
we have stagnation whenever
a(z-c)^{2} + G(z-c) - a^{2}a^{^} = 0 which
at z=d±f
requires G =
(d-c±f)^{-1}
(-a(d-c±f)^{2} + a^{2}a^{^}) .
If z=d-f lies within our z space boundary streamline and z=d+f lies on it
with c = d+f + a(ig)^{↑} then for real a=a we have purely imaginary
G = (-a(ig)^{↑})^{-1}
(-a(-a(ig)^{↑})^{2} + a^{2}a^{^})
= -a(-a(ig)^{↑}) + a^{^}(-ig)^{↑}))
= 2aA sin(g-a)i
and hence G = -4paA sin(g-a) .
For a symmetric aerofoil with g=p we have G=-4paA sin(a) .
Joukowski-Kutta condition flow around a Joukowski aerofoil . | ||
Principle flow F_{+}(Z) |
Reciprocal flow F_{-}(Z) |
Summed flow F_{+}(Z) + F_{-}(Z) |
y_{p} streamlines in green; f_{p} isopotential lines in red; |V_{p}| isospeed lines in blue. |
If one or both of
d±f lie on a boundary C in z space then the warped boundary ¦(C)
in Z space has a gradient discontinuity
at ±2f since
f'(z) = ½(1 - f^{2}z^{-2}) vanishes at z=±f but
f"(z) = f^{2}z^{-3} is ±f^{-1} there .
The reciprocal Joukowski inverse
¦^{-1}_{-}(Z)=½(Z-q) + d = f^{2}(¦^{-1}_{+}(Z)-c)^{-1} + d
has |¦^{-1}_{-}(Z)-d|_{+} = |f^{2}|_{+}|¦^{-1}_{+}(z)-d|_{+}^{-1}
[ Proof :
¦^{-1}_{-}(Z)-d|_{+}^{2} =
(f^{2}(¦^{-1}_{+}(Z)-d)^{-1})^{^}
(f^{2}(¦^{-1}_{+}(Z)-d)^{-1})
= |f^{2}|_{+}^{2} |¦^{-1}_{+}(z)-d|_{+}^{-2}
.]
Consider a d=0 Joukowski warp Z = ¦(z) = z+f^{2}z^{-1} acting on a circle of
centre c=lf
and radius a=(1-l)f chosen so that the circle
passes through z=f and z=(2l-1)f
and also z=(l±(1-l)i)f
while the Joukowski aerofoil passes
through Z=2f and
Z = (2l-1)+(2l-1)^{-1})f
and Z=(l±(1-l)i + (l±(1-l)i)^{-1})f
= (2l±2l(1-l^{2})(l^{2}+(1-l)^{2})^{-1}i)f.
Taking l<0 encloses -f inside the circle and we are left with a boundary gradient discontinuity
with stagnation there if G = -4paA sin(a) =
-4pf(1-l)A sin(a) .
The symmetric aerofoil has length L
= (2-(2l-1)-(2l-1)^{-1})f
= (3-2l+(1-2l)^{-1})f
and width W=4l(1-l^{2})(l^{2}+(1-l)^{2})^{-1}f.
Small negative l yields a thin aerofoil
with L=4(1+l^{2}+2l^{3}+...)f » 4f » 4a
and W » lf with
critical circulation
G=-4paA sin(a) » -pAL sin(a) giving lift
F » 4praA^{2} sin(a) ((a+½p)i)^{↑}
. Letting l®0 gives a flat plate of exact length L=4a and
critical circulation G=-pLA sin(a).
Large negative l yields a "dimpled circular" aerofoil with L » W » 2lf » 2a
and critical circulation G=-2pLA sin(a).
These results appear bizarre, with the sin(a) factor suggesting lift is maximised for an aerofoil moving perpendicularly against the flow,
and the dimpled circle of a diameter L twice as efficient at generating lift as a flat plate of length L.
In practice, the dragless circular flow fails
for attack angles exceeding about 20 ^{o} or nonthin aerofoils, with seperation of the flow from the aerofoil surface
and turbulance in the wake rather than a smooth "reclosing" of the flow behind the aerofoil resulting
in the lift for increasing a, a pheneomena known as stalling.
At the stalling angle, the airflow typically "seperates" from the upper surface and forms unsteady turbulence and eddies. Some authors refer to
stalling as the point where lift becomes negative, but this is erroneous.
Stalling occurs when the still positive lift begins to decrease with increasing attack angle.
For small attack angles the sin(a) factor holds
resulting in lift being roughly linear in a until the stalling angle is approached.
Deriving the critical
Kutta-Joukowski circulation and hence the lift for a general sharp-edged aerofoil
is problematic and various iterative methods such as that of Theodorson have been developed.
Aerofoil Waffle
Loosely speaking, if a is well inside [-½p,+½p] so that the flow is approximately horizontal left to right, and the circulation is negative (clockwise) then
the flow is slower beneath the aerofoil than above it so the pressure s_{0}-½r_{p}V_{p}^{2}
is greater below the aeorfoil than above it and the lift force exerted on the airofoil is vertically upwards
and this is the force that holds up an aircraft. In practice the decrease in pressure above the wing is usually gretaer than the increase in pressure below it
with up to 80% of the lift stemming from "pulling" of the upper surface wing rather than "pushing" of the lower.
Both changes are greater near the leading edge of the wing so that the net torque on the wings serves to
raise the leading edge and we can think of the centre of pressure - being the point on a
designated aerofoil chord line through which we must consider the lift force to act if its couple about the aerofoil centre of mass is to match
the picthing moment - as moving forward. The picthing moment exerted in the wings is typically balanced by the upward lift on horizontal tail fins.
If we lower (or raise) a flap at the trailing edge of an airfoil we will increase drag and slow the
fluid flow below (or above) the airfoil, resulting in an increase (or decrease) in lift.
Thick airfoils, convex or concave, present the problem of lacking a single unambiguous "forward" direction
with reference to which one can define an "attack angle".
The chord line of a (planar section of) an airfoil is the line joining the leading edge
to the tailing edge is sometimes used.
Lift Proportionate to v^{2}
Suppose S is a small planar element
of area content s having unit normal n, and centre of mass c travelling with constant velocity v through a fluid of density r initially
at rest. Assume n is signed so that n¿v ³ 0 .
In small time dt the simplex must displace a small volume s(v^{~}¿n)|v|dt
= s(v¿n)dt
which we assume to act as a single impactive mass we can consider as having
velocity -v and momentum
= rs(v¿n)vdt
on an unmoving simplex.
If the impactive impulse T parallel to n is sufficient to precisely eliminate the n component of the velocity
so that the displaced air "slides off" the simplex we have
T =
rs(v¿n)dt (v¿n)
= rs(v¿n)^{2}dt
which is percived by the simplex as a "drag impulse"
- rs(v¿n)^{2}dt (v^{~}¿n) parallel to v and a "deflective impulse"
- rs(v¿n)^{2}dt (1-(v^{~}¿n)^{2})^{½}
- rs(v¿n)^{2}dt cos(a) where a = ½p - cos^{-1}()(v^{~}¿n)
is known as the angle of attack in the context of a thin planar airfoil.
This is a gross simplification of airfoil dynamics but substantiates somewhat
our assertion that the forces acting
on an airfoil are roughly proportionate to the square of the airspeed as well as to the area of
the "leading" surface.
Force Velocity Coefficients
Broadly speaking (and for subsonic speeds), the lift and drag are individually proportional to v^{2} , the density of the medium, and the
(XY planar) area of the airfoil and this is usually expressed for N_{0}=3 as
L = ½rSv^{2} C_{L,a,b} ;
D = ½rSv^{2} C_{D,a,b} ;
C = ½rSv^{2} C_{C,a,b}
where r is airdensity, S is wing area and C_{L,a,b}, C_{D,a,b}
, C_{C,a,b}
are known as lift coefficient , drag coefficent, and crosswind coefficient
for attack angle a=q-½p and yaw angle b=f for the airfoil orientated with e_{3} upwards
amd e_{1} forward. These are the projections of T into the wind frame , with D parallel to v ,
L lieing along ¯_{v*}(e_{3}_{Q}) , and (moving to 3D) C perpendicular to both.
For a (Z) symmetric airfoil , C_{L,0,0} = 0 while C_{D,0,0} ³ 0 since v is here the velocity of the fluid past the aerofoil rather than the velocity of the aerofoil trhough the fluid. Modern airfoils are asymmetrically cambered to give
small C_{L,0,0} > 0. The crosswind force is frequently neglected, or assumed cancelled by the opposing crosswind force on a symmetricaslly
opposing aerofoil.
For inviscid irrotational steady flow we have
C_{D,a,b}=0 ; C_{L,a+p,b}= C_{L,a,b} ;
C_{C,a+p,b}=- C_{C,a,b} but in practice we usually have
aerofoils less efficient when travelling backwards with
C_{D,a+p,b} > C_{D,a,b} > 0 and C_{L,a+p,b} £ C_{L,a,b}
Aerofoils are frequently characterised by providing C_{L,a} º C_{L,a,0},
C_{D,a}, and C_{M,a,} for a limited range of a typical in
normal flight, say -4 to 20 degrees
though the values
R_{D L}(a)=
( C_{L,a}^{2}+ C_{D,a}^{2})^{½}
and Q_{D L}(a)= tan^{-1}( C_{L,a}/ C_{D,a})
are arguably more "natural"
since for N_{0}=2 we have
F
= ½rSv^{2}R_{D L}(a)(Q_{D L}e_{21})^{↑}v^{~}
= ½rS|v|R_{D L}(a)(Q_{D L}e_{21})^{↑}v
with
Q_{D L}(a)=±½p for dragless lift.
Sometimes C_{D,a}=(pEA)^{-1} C_{L,a}^{2} is assumed where efficiency E»1
and A is a aspect ratio.
For aerobatic emulation purposes we typically require values for all a and b.
For large a the aerofoil can sometimes
be considered a flat plate, although as the dragless circulatory flow model fails for large a for flat
plates an alternate high drag model model should be employed.
C_{L,a} is small or zero for zero a, rising roughly linearly to a maximum
at a stalling angle of about 16 degrees, and then falling off in an uncertain manner,
possibly to zero value and zero gradient at ½p.
For aÎ(½p,p]
we might crudely assume
C_{L,a+p}=m_{L} C_{L,a} where 0<m_{L}£1 is a reversal inefficiency factor.
and characterise or tabulate C_{L,a} only over [-½p,½p) .
The assumption that the reversal lift profile has the same shape as the forward one is unrealistic but if we expect
an aerofoil to seldom travel backwards it may be acceptable.
For aerofoils symmetric in their e_{1}_{W} chordline, the lift for attack angle -a is minus that due to attack angle a.
One possible strategy is to assume that for b Î [0,½p] C_{L,a,±b} = cos(b)^{2} C_{L,a,0}+ sin(b)^{2} C_{L,a,±½p} and tabulate forwards and sideways lift coefficients C_{L,a,0} and C_{L,a,±½p} over aÎ[-½p,½p) , with C_{D,a,b}, C_{D,a,b} and the moment coefficients implemented similarly. The use of cos(b)^{2} and sin(b)^{2} weights ensures that C_{L,a,b}= C_{L,a,0} for a solid of revelution about e_{3}_{W} disk like aerofoil.
Sometimes one or more of the ½, r and S
are amalgamated into the coefficients ,
although r does decrease with altitude and varies across shock waves. However, keeping the density r and area S outside the coffecicients
keeps them dimensionless, dependant only on the shape rather than the size of the aerofoil.
Torque Velocity Coefficients
The point of a 2D airfoil through which the lift and drag forces can be considered as acting (for a given a),
known as the centre of pressure is also relevant, for unless this happens
to be the centre of mass of the airfoil, the forces will indice a torque known as a pitching moment tending to rotate the airfoil.
This torque is often written as ½rScv^{2} C_{M,a}p) where C_{M,a}p) is the pitching moment coefficient
for attack angle a and point p, about which the torque is computed. c is the chord length.
We will again embody the ½rS and now c also into the coefficient and obtain
h_{p} = v^{2} C_{M}(a,p) for the picthing moment about a Y-eaxis through point p.
For many airfoils there is a point a (more accurately an axis) known as
the aerodynamic centre about which the pitching moment for a given airspeed v^{2} is largely invarient with a
(at least over a typical flight range of a of -8 to 16 degrees) so that the moment coefficient C_{M,a}a) is independant of a.
This point is usally about ¼ chordlength back from the leading edge of the airfoil, and the constant torque tends
to be signed so as to lower the lower nose and raise the tail of an aircraft (or twist the wings from the fuselage!)
The pitching moment tending to rotate the aerofoil is usually taken to be
½rV_{p}^{2}ScC_{m}(a,b) where c is a mean chord length
and C_{m}(a,b) is a dimensionless coefficient.
Similarly for a 3D aerofoil we have rolling and yawing torques about the v^{~} "wind upward"
axies exerted on the aerleron due to the fluid drift v.
Bimpulses due to spin
Even under our assumption that the fluid velocity is uniform in the vicinity of the body, a nonzero
spin w of the body means that the fluid encountered by the body surface will have varying relative velocity.
If an aircraft is rolling anticlockwise about e_{1}_{Q} so that its left wing is rising, for example, then
the effective angle of attack a of the airstream over the left wing is decreased at a point l out along the wing
by roughly lw^{23}|v|^{-1}
while that on the right wing is increased by about the same ammount.
Assuming that all the lift ½rv^{2}C_{L} derives from the wings, these
changes in a result in lift decreasing and increasing by
½r_dlcv^{2} lw^{23}|v|^{-1} ¶C_{L}/¶a
on the left and right wings respectively, imparting a clockwise torque
r|v_dlcl^{2}w^{23}¶C_{L}/¶a
acting in conjunction with and probably exceeding any "fractional" torque proportionate to
(w^{23})^{2}
opposing the roll.
We can regard this as providing an addition
½l|v|^{-1}w^{23}¶C_{L}/¶a to the roll coefficient
C^{23}(v^{~})
= C^{23}(a,b) providing the characteristic length l used for the effective wing length
is the same as the length used to render the roll coefficient dimensionless.
Similarly, an anticlockwise yaw about e_{3}_{Q} bringing the right wing forwards will,
if the aircraft is travelling horizontally with nose slightly raised, tend to decrease the right wing attack angle but increase its effective speed
causing a net increase in lift on the right wing and a decrease on the left, resulting in a torque inducing a roll
lowering the wing on the "inside" of the yaw. This can be crudely modelled by a further addition
½l|v|^{-1}w^{12}¶C_{L}/¶b to the roll coefficient
and in general we model the biimpulses due to spin by adding such terms involving a and b derivatives of the velocity based force
coeefficients to the ½N_{0}(N_{0}+1) velocity response coefficients.
Consider a long wide aerofoil of stance 1 travelling with
velocity v=(-ae_{31})^{↑}e_{1} » e_{1}-ae_{3} for small a and spinning
at w. A segment at le_{2} has velocity v+le_{2}Ùw.
Let us first consider yaw rotatiaon with w=we_{12} with w>0. The segment velocity is then_v-lwe_{1}
so for l>0 the speed decreases but the attack angle increases unless a=0.
Matters thus simplify if a=0 since then the attack angle remans constant and the speed becomes
v-lw so that the total lift is
F = ò_{-L}^{L} dl½rc(v-lw)^{2} C_{L,0} e_{3}
= ½rS C_{L,0}(v^{2} + 6^{-1}(2L)^{2}w^{2})e_{3}
where L is the demiwingspan and c=(2L)^{-1}S approximates the chord length for a thin aeorfoil.
The total roll torque is
ò_{-L}^{L} dl½rcl(v-lw)^{2} C_{L,0} e_{23}
= -12^{-1}rS C_{L,0}v(2L)^{2}we_{23}
equivalent to adding -6^{-1} C_{L,0}v^{-1}2Lw to the velocity based roll moment coefficient.
For nonzero a but zero b we have sin(Da)(lw)^{-1} = sina(v+Dv)^{-1}
F = ò_{-L}^{L} dl ½rc((v cos(a)-lw)^{2}+v^{2} sin(a)^{2}) C_{L, tan-1( sin(a)( cos(a)-v-1lw)-1)} , typically with |lw|<<v .
This is the model used by many flight simulators. The six (½N_{0}(N_{0}+1) for N_{0}=3) velocity induced wind frame force and torque coefficients
for the whole aircraft are tabulated, often with thousands of data points perhaps
clustered more densely over more common regions , over the encountered ranges
of flight (wind) angles a,b and mach speed M.
Effects due to aircraft spin, acceleration, propellor thrust, flap settings and so forth are
the implimented as approximations based on the velocity coefficients such as the addition of
½lw^{23}|v|^{-1} ¶C_{L}/¶a
to the roll moment velocity coefficient, obtainable via a calculated differential of the tabulated C_{L,a,b}M) surface.
It is hard to adapt this whole-body model to accomodate damage to particular regions. If a rear tail fin is half shot off, for example, then we lack LUTs for the new assymetric composite aircraft.
For the emulation of near level flight when velocity is largely forward with speed sufficient for the lift to roughly balance the gravitational weight we can
safely extract the V_{p}^{2} and parameterise the coefficients solely by airspeed direction but for acrobatic flights
involving high spins and accelerations our coeffecients become functions of w as well as v
and any control parameters.
We can extend our
½N_{0}(N_{0}+1) boy frame coefficients as
T = ½r|W|^{2}Så_{i<j}l_{i}
C^{ij}(W^{~},m,M)e_{ij¥}
where W is the bivelocity of magnitude |W| and M is the mach speed of v.
The advantage of moving the v^{2} outside the velocity coefficients where twofold. First, it makes the coefficient
dimensionless, and second it makes it a function of only the direction v^{~}. This second advantage is voided if the mach speed M
is reintroduced as a parameter so we may as well declare
T = ½r|W|^{2}Så_{i<j}l_{i}
C^{ij}(W,m)e_{ij¥} with dimensionless coefficients that are functions of the unnormalised ½N_{0}(N_{0}+1) coordinate bivector W
and a control vector m.
Propellers
The principle example of a nonfixed aerofoil subbody is a propellor blade. These will typically be mounted at symmetrically
on an aircraft with e_{3}_{P} = e_{1}_{Q} and be spun rapidly about this axis by an engine
to provide a forward thrust usually adequate only for subsonic flight, or lift in the case of a helicopter.
The path taken by a point on the propellor is a helix, skew if v_{PQ} ¹ e_{1}
and curved if w_{Q}¹0.
The instantaneous force exerted on the propellor
has a "thrust" component along e_{1}_{_APlane} and a "torque" conmpoent perpendicur to it
which is typically balanced by and opposite torques from
the opposite arm of a two-blade propellor, or averages out over a propellor rotation.
The net force exerted on the propellor centre is typically parallel to e_{1}_{_APlane} but induces a troque on the aircraft if the
propellor centre c_{PQ} lies off the centre line of the aircraft. Propellors are typically symmetrically or centrally mounted
but may be above or below the aircraft mass centre and so impart a pitching moment.
Propellors accelerate the air passing "through" them and typically increase the airspeed over the wings and/or tail and hence lift.
They also act as gyroscopes resisting pitch and yaw torques.
Consider a point at -ye_{2} of a propellor having stance 1 and bivelocity
W=(1+½v_e3e)w_e12_frm2(P,Q).
Its velocity is ywe_{1}+ve_{3} which we can regard as a 2D areofoil travelling at speed
(y^{2}w^{2}+v^{2})^{½} with attack angle g(y)-q(y)
where angle g(y) is the pitch of the propellor blade at distance y from its hub
and q(y)= tan^{-1}(v(yw)^{-1}). The force along e_{3} acting on the 2D section
is
dF =
½rdS(y)(y^{2}w^{2}+v^{2})(
C_{L}(g(y)-q(y)) cos(q(y)
-C_{D}(g(y)-q(y)) sin(q(y))
=
½rdS(y)
R_{D L}(g(y)-q(y)) sin
(g(y)-q(y)) cos(q(y)
- Q_{D L}(g(y)-q(y)) )
where dS(y)=dy L(y) is the propellor area element. Integrating this over y from 0 to the propellor span
gives the instantaneous forward (e_{3}) force exerted by one blade of the propellor.
However, and easier way to estiamte the total force T=Te_{3} exerted by a propellor on its mounting point
is to assume that the propollor has a given efficiency EÎ[0,1] so that
if the engine exerts a torque Q and so does Qw work then EQw = T_vu3
where _vu3 is the forward speed of the aircraft, so that T=(_vu3)^{-1}Qw.
Body Frame Coefficients
Returning briefly to 2D, our notion of fluid drift v=A(ai)^{↑}e_{1}
travelling for small attack angle a from left to right suggests an aerofoil W aligned with its forward direction e_{1}_{W}=-e_{1} facing right to left.
More generally, we might expect an aircraft Q to be pointing roughly into the enocuntered windpeed with v^{~} » -e_{1}_{Q} .
If we wish to work in the frame of the aerofoil it is sensibkle to take v to be negative the encountered fluide velocity
so that directly forward motion of the aireofoil corresponds to a=b=0
More generally we can define
N_{0}+½N_{0}(N_{0}-1)=½N_{0}(N_{0}+1) dimensionless coefficients defined by
expressing the 3-vector bimpulse
T = ½rv^{2}Så_{i<j}l_{i}
C^{ij}(v^{~})e_{ij¥}
in the frame of the body rather than the windframe. Here i,j are from {0,1,..,N_{0}} , l_{i} is a characteristic length of the aerofoil in the i dimension with
l_{0}=1, and
C^{ij}(v^{~}) is a dimensionless scalar coefficient.
C^{01}(v^{~}) is the forward coefficient with
C^{01}(p-a,0) = - cos(a) C_{D,a,0}+ sin(a) C_{L,a,0} ;
C^{02}(v^{~}) is the sideways coefficient ;
C^{03}(v^{~}) the upward coefficiecnt with
C^{03}(p-a,0) =
cos(a) C_{L,a,0}+ sin(a) C_{D,a,0} ;
C^{13}(v^{~}) the body pitching coefficient
with C^{13}(p-a,0)= C_{M,a,0}
, and so on.
In 3D we can approximate these using
C^{ij}(a,±b)
= cos(b)^{2}C^{ij}(a,0) + sin(b)^{2}C^{ij}(a,±½p) as for the windframe coefficients
and because we are now measuring the coefficients in the body frame, for sideways symmetric 3D
body symmetric in y=0 we have
C^{ij}(a,-b)=±C^{ij}(a,b) with the - occuring when one of i or j is 2.
Speed of Sound
Compressable fluids can carry density waves
Even though in theory the velocity based flight coefficients depend only on the direction V_{p}^{~} and not on the magnitude V_{p}^{2}, in practice they do vary with the speed partcularly when distingusihing slow, subsonic, and supersonic flight.
It is natural to express the airspeed |v| as a fraction of the speed of sound s=(r^{-1}m)^{½}
where fluid stiffness coefficient m has units kg m^{-1} s^{-2}, ie. we set
|v| = M(r^{-1}m)^{½} for unitless mach number or mach speed M so that dymanic pressure
½rv^{2} = ½M^{2}m is independant of the density r; or rather
its dependance on r is subsumed into M
. For M>1 we have supersonic speed
and M>5 as hypersonic speed .
The speed of sound varies with density and tempreature and the nature of the fluid but for air at sea level
it is roughly 2^{8.4} ms^{-1}
Furthermore, aircraft typically have control surfaces like aerlerons and flaps whose orientations are varied to
effectively alter the flight coefficients of the wings or fins to which they are attached.
The settings of such control surfaces are typically parameterised by angles and we have a
control vector m fromed from these angles.
Body as rigid amalgamation of simpler subbodies
An aletrnative approach is to consider the aircraft as composed of a collection of rigidly connected aerofoils, each with their own aerodynamical coefficients.
We might for example consider the left wing to be composed of one or more wingsections.
Assuming a particular wing section W to be at rest with respect to the aircraft, the instantaneous apparent bivelocity of the fluid as percieved by W is
-W_{WW} =
-R_{W4}^{§}_{§}(c_{Q4}dt)e_{¥}
- R_{WQ}^{§}(
(c_{WQ}¿w_{QQ} )e_{¥}
+ w_{QQ}
)R_{WQ}
and we can use this to construct the biimpulse about the W centre
T_{W}(-W_{WW})
applied to the wing section in the frame of the wing section and then recast this as
R_{WQ}_{§}( T_{W}(-W_{WW}) ) into biimpulse about
c_{WQ} applied to the aircraft,
assuming the imulses and tiorques involved are inadequate to deform or dislodge the wing section.
The biimpulse about the aircraft centre in the aircraft frame is
R_{WQ}_{§}( T_{W}(-W_{WW}) )
+ c_{WQ}Ù(
e_{¥0}¿
R_{WQ}_{§}( T_{W}(-W_{WW}) ) )Ùe_{¥} and by adding
such centred biimpulses for all the aircraft subbodies we obtain a net biimpulse T_{c} acting on the aircraft
about its centre c which we typically take to be be the mass centre.
The kinematical responce of the aircraft is then given by
DW = H^{-1}_{Q}(T_{c}).
This method is again highly artificial. Even in steady, level flight, the airflow encountered by
tail is typically in the downward deflected downwash from the wings, and so has its attack angle lessened.
In accelerative motion, vortices or similar flow effects shed from the wings hit the tail planes after a delay roughly proprtionate to
|v|^{-1}, a crosswind coming from the left is partially screened from the rightwing by the fuselage,
and in general the fluid flow in the vicinity of the body is extremely complicated. However, modelling the fluid itself
is typically computationally out of the question and the best we can hope for is broadly plausable behaviour
of the body moving through it.
Leftovers
Problems with 1-flows
Imagine a medium consiting of two types of particle. "Green" particles of small unit "mass" and unit positive "charge",
and "red" particles of unit mass and negative charge which
with regard to a partcular observer e_{4} , are confined to a ring of radius R centred on the origin,.
Assume that there are equal large quantities of green and red particles
distributed with uniform density r particles per (small) unit volume around the ring
and that particles contrive to avoid impacting eachother, or do so only rarely. Assume
that the green partciles are travelling "anticlockwise" around
the ring with orbital period T while the red particles travel clockwise with the same orbital period.
The net spacial flow of mass through a given cross sectional ring element is then M_{p}=0,
although there is kinetic energy
2r½(2pR/T)^{2} = r(2pR/T)^{2} .
The net spacial current flow is 2 r(2pR/T) anticlockwise . If the red particles were all to reverse direction, and flow counterclockwise,
we would have zero current
but nonzero momentum.
This simple example demonstrates the complications in "adding up" flows. Allowing negative and postive charges
means that the sum of two charge-weighted timelike vectors may not remain timelike.
Generally speaking, when adding four-momentum 1-vectors, relative masses combine additively but
relative three-momentum magnitudes incurr a triangle inequality so that proper magnitude is not conseerved
and tends to increase .
When physicists speak of the flow "at" a particular point they usually mean the averaged flow over a small volume at that point, and by "small"
they usually actually mean large enough to contain multiple particles. They hope by summing a large
number of fundamentally discontinous point-dependant "path localised" functions to produce a continuos field.
Thus all the matter at a given event p is considered to be effectively flowing with the same direction and speed.
We will refer to such a flow as single flow because all the matter at p is effectively doing the same thing.
More generally we represent a superposition of 1-flows with a 2-tensor v_{p} such that
v_{p}(l,d)_{aÙb} provides the number of particles of the matter at p of
kind classified by a matter-type indicator l
flowing across spacial 2-plane aÙb in e_{4}-time d. By which we mean
that on fixing any timelike e_{4} perpendicular to a and b the integer number of particles crossing
between e_{4}-times t and t+d divided by d.
In the sense of v_{p} representing the documentation of an actual multiparticle flow,
we regard v_{p}(l,d)_{aÙb} to be valued in integer multipes of d^{-1}
rather than a full real type.
As d®0 the integer multipliers get smaller discontinuously.
If the flow is steady in the sense that at a suffiicently small d the second derivatives
become swamped by the first and we can conceptually replace individual particles with
shoals of identical smaller particles having parallel trajectory
then we expect v_{p}(l,d)_{aÙb} to approach a limit .
Flow is thus more naturally regarded as a bivector which we will call 2-flow. For N=3, the 2-flow is dual to the 1-flow .
Flow rond moving sphere
Bernouilli's equation concerns a sphere moving through a fluid and states that at anypoint near the surface of a frictionless
sphere whose centre moves with low velocity v Î e_{12} in the x^{3}=0 plane is
sg^{-1} + x^{3} + ½v^{2}g^{-1} = k
where s is the fluid pressure , g is gravitational accelaration, and k is a constant.
Thus there is a symmetry to the flow and the fluid is left essentially undisturbed, no energy is lost,
and drag is zero.
If the velocity, or friction, increases, there can instead be eddies or turbulance in the "wake" of the sphere.
imparting energy to the fluid and so slowing the sphere;s travell.
In practice the "impactive drag" or form drag on a sphere is about half that on a flat plane of
identical leading area travelling
"flat on" (a=½p , v^{~}=n) . Optimal streamlining can reduce this drag to about 5%.
We can rearrange Bernoulli as
sr^{-1} + ½v^{2} + x^{3}g = kg and if the sphere is small so that
x^{3}g may be neglected we have
s + ½rv^{2} = kgr . If the fluid is considered as incompressible
so that r is constant we refer to constant s_{0} º kgr
as that stagnation pressure since it correspends to P when v=0,
and ½rv^{2} = s_{<0>}-s as the dynamic pressure.
Hence the greater |v|, the less s , provided |v| remains small enough (less than half sonic speed)
for the Bernouilli equation to approximately hold .
Extremal M-Curves
A standard problem in classical mechanics is to determine the form of the curve assumed by a
rope of length a hanging motionless between two fixed points a distance less than a apart
under a uniform gravitational field, which is an extremal 1-curve in N=2 dimensions.
If we assume the rope to hold a curve shape p(s)=(x(s),y(x))
(the Y axis being gravitationally vertical) that does not loop "over" itself vertically, we have
^{'}^{2} = dx^{2} + dy^{2} so that the length constraint is
a = ò_{0}^{1} ds = ò_{x0}^{x1} dx(1+y'^{2})^{½}
where y' º dy/dx.
The mechanical constraint is that total gravitational potential energy
ò_{0}^{1} ds rgy(s) = rg ò_{x0}^{x1} dx y(1+y'^{2})^{½}
be minimised,
where g is uniform vertical gravitational acceleration and r is the mass of unit length of rope.
More generally, we might seek to minimise
ò_{x0}^{x1} dx¦(x,y,y') subject to a constraint
ò_{x0}^{x1} dxg(x,y,y') = a.
A standard approach is to form h(x,y,y') º ¦(x,y,y') + lg(x,y,y')
where scalar l is known as a Lagrange multiplier. One then embodies freedom to vary the
path by means of two
z_{0} and z_{1} such that
y(x_{0},z_{0},z_{1})=y(x_{0}) ; y(x_{1},z_{0},z_{1})=y(x_{1}) " z_{0},z_{1} (boundary condition)
; y(x,0,0)=y(x) ; and y(x,z_{0},z_{1}) is twice continuosuly differentiable in all parameters.
Forming K(z_{0},z_{1}) = ò_{+x0}^{x1} dx h(x,y(x,z_{0},z_{1}),y'(x,z_{0},z_{1})) the extremal condition requires
¶K/¶z_{0} and ¶K/¶z_{1} to vanish at z_{0}=z_{1}=0 and this leads (via integration by parts)
to the Euler-Lagrange equation
¶h/¶y = d/dx(¶h/¶y') .
In the case of the hanging rope,
h(x,y,y') = (rgy-l)(1+y'^{2})^{½} and setting l = -rgy_{0} we have
h(x,y,y') = rg(y-y_{0})(1+y'^{2})^{½}
= rg(y-y_{0})(1+(y-y_{0})'^{2})^{½} for some constant y_{0} .
Setting h = y-y_{0} the Euler-Lagrange equation is
rg(1+h'^{2})^{½} =
d/dx rghh'(1+h'^{2})^{-½}
with first order simplification
h' rghh'(1+(y-y_{0})'^{2})^{-½} -
rgh(1+h'^{2})^{½} = b
Þ
rg(1+h'^{2})^{-½}h = b
having solution h = b cosh(x-c)/b) so the rope has shape
y_{0} + b cosh((x-c)/b) where c,y_{0} and b are chosen to
match the given endpoints.
We say an M-curve C_{M} is M-extremal for an action functional
¦_{CM}:U^{N} ® U_{N} if some particular magnitude measure (content, scalar part, square, modulus or whatever) of
ò_{CM} d^{M-1}p ¦_{CM}(p) is maximised (or minimised) by C_{M} in the sense that
integrating other any M-curve which deviates only slightly from C_{M} will produce
a result no higher (or lower) than the C_{M} integral. C_{M} is a "locally optimal"
M-curve for a particular integration ¦_{CM}. We specify the dependancy of ¦ on C_{M} to allow
action functionals dependant on the geometric properties of the "sampling curve" C_{M} , such as tangent
or normal vectors , to allow "direction dependant sampling" or "velocity dependance".
Euler-Lagrange Equations
For M=1 we have a path p(s) 1-extremal for F_{C1} = F(s,p(s),p'(s)).
Hamilton's principle provides that the state of a system characterised at time t by k multivector variables
x_{1}(t),..,x_{k}(t) varies from time t_{0} to t_{0} so as to 1-extremise
a (usually real scalar) integral measure
S(t) º S(t_{0}) + ò_{t0}^{t0} dt L(t,x_{1}(t),..,x_{k}(t),xdt(t),...,x_{k}dt(t))
with the action functional L(t,x_{1}(t),..,x_{1}dt(t)) known as the Lagrangian
of the system.
L is usually assumed to be real-scalar valued and independant of p"(t) and higher derivatives and is
traditionally assumed to seperate into "kinetic" and "potential" componenets independent of time and velocity respectively
as
L(t,p(t),p'(t)) = T(p(t),p'(t)) - V(t,p(t)) but more generally we might postulate
a multivector-valued Lagrangian of k multivector-valued variables and their first temporal derivatives
L(t,x_{1}(t),..,x_{k}(t),x_{1}dt(t),..,x_{k}dt(t)).
If the Lagrangian L is itself an integral over some spacial M-curve (typically an
M=N-1 hypercurve reprenting a contemporal slice of a B_{ase} space)
L = ò_{Base(t)} d^{N-1}p L(t,p,pdt)
that spacially integrated function is known as a Lagrangian density. Note that such requires a "velocity" pdt be associated with every point p in B_{ase}(t) .
Extremal Paths
A standard approach for M=N=1 to 1-extremise ¦(s,x,x')
is to embody freedom to vary the path by means of two scalar parameters
z_{0} and z_{1} such that
x(s_{0},z_{0},z_{1})=x(s_{0}) ; x(s_{1},z_{0},z_{1})=x(s_{1}) " z_{0},z_{1} (boundary condition)
; x(s,0,0)=x(s) ; and x(s,z_{0},z_{1}) is twice continuosuly differentiable in all parameters.
Forming K(z_{0},z_{1}) = ò_{s0}^{s1} ds¦(s,x(s,z_{0},z_{1}),x'(s,z_{0},z_{1})) the extremal
condition requires
¶K/¶z_{0} and ¶K/¶z_{1} to vanish at z_{0}=z_{1}=0 and this leads (via integration by parts)
to the Euler-Lagrange equation
¶¦/¶x = (d/ds)(¶¦/¶x') .
In particular cases where ¦(s,x,x') = ¦(x,x') so that ¶¦/¶s=0 and there is no explicit s
dependence,
we have d/ds(x'¶¦/¶x' - ¦)= 0
giving a first order differential equation
x'¶¦/¶x' - h = constant .
[ Proof : y" ¶h/¶y' + y'd/dx¶h/¶y' - ¶h/¶x-¶h/¶yy' - ¶h/¶y'y"
= y'(d/dx¶h/¶y' - ¶h/¶y) - ¶h/¶x
= 0
.]
A nongeometric generalisation is to extremise ò_{s0}^{s1} ds¦(s,x_{1},x_{2},..x_{k},x_{1}',x_{2}',..x_{k}')
subject to M scalar constraints g_{i}(s,x_{1},x_{2},...x_{k})=0 i=1,2,..,M
(Note the absence of any g dependance on the x_{i}').
To do so we form
h(s,x_{1},..,_xK,x_{1}',..,x_{k}') = ¦(s,x_{1},..,_xK,x_{1}',..,x_{k}')
+ å_{j=1}^{M} l_{j}(s)g_{j}(s,x_{1},..,x_{k},x_{1}',..,x_{k}')
where l_{j}(s) are M arbitary functions of s
and obtain Euler-Lagrange formulae
¶h/¶x_{i} = (d/ds)(¶h/¶x_{i}') for i=1,2,..k .
Generalising geometrically, we have F(s,x_{1},x_{2},..,x_{k},x_{1}',x_{2}',..,+x_{k}')
with geometric Euler-Lagrange Equations
¶F/¶x_{i} = (d/ds)(¶F/¶x_{i}') " i=1,2,..k
and first order equation
å_{i=1}^{k} (x_{i}'_{*}Ñ_{xi'})F - F = constant (independant of s)
if ¶F/¶s = 0.
Considering the x_{i} as seperate grades of a single multivector argument x we can regard the Euler-Lagrange equations
as individual coordinate terms of a single geometric equation
¶_{x}F(s,x,x') = (d/ds)(¶_{x'}F(s,x,x'))
where ¶_{x} = å_{ijk..} e^{ijk..}¶/¶x^{ijk..}
over all blades comprising x space so that, for example,
¶_{x} = å_{i=1}^{N} e^{i}(¶/¶x^{i}).
Generalised Momentum
When F is a scalar-valued Lagrangian we have (x_{i}'_{*}¶_{xi'})L =
x_{i}'_{*}(¶_{xi'}L) and multivector m = Ñ_{xi'}L(x_{1},..x_{k},x_{1}',..,x_{k})
is known as generalised spacial momenta or cononical momenta.
Typically x_{i}' is spacial 1-vector valued and so momenutum is a spacial 1-vector.
If multivector L is independant of x_{i} so that ¶L/¶x_{i}=0 then the i^{th} Euler-Lagrange equation provides
(d/dt) ¶L/¶x_{i}' = 0 so M_{i} = ¶L/¶x_{i}' is constant, ie. unchanging with t,
Thus the constants of a system are consequences of absent dependendencies (aka. symmetries) in the Lagrangian.
Energy is conserved when L depends on t only indirectly via x(t) and it derivatives.
Momentum is conserved when L depends on x only indirectly via x'.
For L(t,x,x') = ½mx'^{2} - f(x,t) we obtain classical momentum
Ñ_{x'}½mx'^{2} = mx' , constant if f(x,t)=f(t).
Adding an electromagnetic term qc^{-1}a_{x}¿x' to L introduces
qc^{-1}a_{x} to the momentum
If
L(t,x_{1}(t),..,x_{k}(t),x_{1}dt(t),..,x_{k}dt(t))
= L(x_{1}(t),..,x_{k}(t),x_{1}dt(t),..,x_{k}dt(t))
so that there is no explicit t dependance then
the Hamiltonian H(x_{1}(t),..,x_{k}(t),x_{1}dt(t),..,x_{k}dt(t))
º å_{i=1}^{k} (x_{i}dt_{*}Ñ_{xidt})L - L is a constant,
the "energy" of the system.
We can regard the Hamiltonian as generalised temporal momentum. Conservation of energy but varying spacial momentum
resulting from only non-relativistic potentials f(x,t)=f(x) .
But H = - ¶S/¶t .
Even though, like the Lagrangian, S(t) cannot strictly be regarded as a function of position since it is only defined over
our given extremal path, we can nontheless postualte an action field S(t,x) an it can be shown that
m = (x_{i}'_{*}¶_{xi'})L = Ñ_{xi}S(t,x_{i}) and in particular we have the Hamilton-Jacobi equation
H(x_{i},x_{i}',t) = -¶S/¶t
[ Proof : For any ¦(x,x')
d/dt ( å_{j=1}^{k} x_{i}' ¶¦/¶x_{j}' - ¦)
= å_{j=1}^{k} x_{j}" ¶¦/¶x_{j}'
+ å_{i=1}^{k} x_{j}' d/dt ¶¦/¶x_{j}'
- ¶¦/¶t
- å_{j=1}^{k} ¶¦/¶x_{j}x_{j}'
- å_{j=1}^{k} ¶¦/¶x_{j}'x_{j}"
=
å_{j=1}^{k} x_{j}' d/dt ¶¦/¶x_{j}'
- ¶¦/¶t
- å_{j=1}^{k} ¶¦/¶x_{j}x_{j}'
=
å_{j=1}^{k} x_{i}'( d/dt(¶¦/¶x_{j}') - ¶¦/¶x_{j} )
- ¶¦/¶t
=
å_{j=1}^{k} x_{j}'( d/dt(¶¦/¶x_{j}') - ¶¦/¶x_{j} )
if ¶¦/¶t = 0
= 0
if d/dt(¶¦/¶x_{j}') = ¶¦/¶x_{j} j=1,2,..,k
which are the Euler-Lagrange equations.
d/dx ( å_{i=1}^{k} y_{i}' ¶¦/¶y_{i}' - ¦)
= å_{i=1}^{k} y_{i}" ¶¦/¶y_{i}'
+ å_{i=1}^{k} y_{i}' d/dx ¶¦/¶y_{i}'
- ¶¦/¶x
- å_{i=1}^{k} ¶¦/¶y_{i}y_{i}'
- å_{i=1}^{k} ¶¦/¶y_{i}'y_{i}"
=
å_{i=1}^{k} y_{i}' d/dx ¶¦/¶y_{i}'
- ¶¦/¶x
- å_{i=1}^{k} ¶¦/¶y_{i}y_{i}'
=
å_{i=1}^{k} y_{i}'( d/dx(¶¦/¶y_{i}') - ¶¦/¶y_{i} )
- ¶¦/¶x
=
å_{i=1}^{k} y_{i}'( d/dx(¶¦/¶y_{i}') - ¶¦/¶y_{i} )
if ¶¦/¶x = 0
= 0
if d/dx(¶¦/¶y_{i}') = ¶¦/¶y_{i} i=1,2,..,k
which are the Euler-Lagrange equations.
.]
Theoretical physics then becomes a quest for the One True Action Lagrangian density , usually assumed
real scalar and generating kinematic equations involving zeroth, first, and second differentials only.
Geometric (multivector-valued) Lagrangians extermised in the sense that each multivector coordinate is stationary under variation.
This is an intrinsically nonrelativistic approach in the case of multiple particulate systems since it requires
a favoured temporal parametrisation t with L(t,p,p') = L(p,p')
As "locally shortest routes", Geodesics are 1-extremals for the scalar path length
of ò_{0}^{t} ds|dp¿g(dp)|^{½} . Integrating a square root can be messy
but fortunately such paths also extremise
ò_{0}^{t} ds ds p'(s)¿g_{p(s))}(p'(s))
where p'(s) denotes the (d/ds)p(s)
. _Be Thus we set
L(p,p',s) =
p'¿g_{p}(p')|^{½} and
minimise scalar integral ò_{0}^{t} ds ds L(p(s),p'(s),s) by solving the N Euler-Lagrange equations
d/ds(¶L/¶p^{i}') =
¶L/¶p^{i}) for i=1,2,...N .
L(p,p',s) will generally depend on p via _gp but if _gp has a symmetry such that
¶L/¶p^{i}=0 for a particular coordinate i then
¶L/¶p^{i}' is constant along any geodesic path.
Geodesic flow can consequently be viewed
as arising from a Lagrangian density
L(p(t),p'(t)) = |p'(t)¿_gp(p'(t))|^{½} .
with particles following timelike trajectories that extremise (minimally) their proper time (arclength).
If t is proper time parameterisation then
L(p(t),p'(t)) = |p'(t)^{2}|^{½} = 1 along a geodesic.
Electrodynamical forces due to a 1-vector four-potential a_{p} (typically with Ñ_{p}¿a_{p}=0) are introduced by adding scalar
-q p'(t)¿ a_{p(t)}
to the Lagrangian density leading to the Lorentz force law
mp"(t) = qf_{p}.p'(t)
where f_{p} = Ñ_{p}Ùa_{p} .
[ Proof :
L = m(-p'^{2})^{½} - q p'¿a_{p}
Þ Ñ_{p}L =
Ñ_{p}(m(-p'^{2})^{½} - q p'¿ a_{p})
= -qÑ_{p}(p'¿a_{p}) ;
Ñ_{p'}L =
-m(-p'(t)^{2})^{-½} p'(t) - qa_{p}
so the Euler-Lagrange equation is
qp'¿Ñ_{p}a_{p} =
(d/dt)(m(-p'^{2})^{-½} p' + qa_{p} )
= mp" + q(p'¿Ñ_{p})a_{p}
Þ mp" = q(Ñ_{p}(a_{p}¿p') - (p'¿Ñ_{p})a_{p}))
= q(Ñ_{p}Ùa_{p})¿p'
.]
Extremal surfaces
A more geometric generalistaion is to consider the problem of finding the M-curve of given content
maximising a particular boundary or interior integral. We might, for example, seek the loop
starting and ending at a given point a and constrained to lie in a given 2-curve (surface) containing a that maximises
enclosed content (area) for a given boundary content (pathlength)
Of fundamental importance is the fact that spacial Lagrangian density
(Ñ_{[e123]}y_{p})^{2} is extremised for integration over C_{M}
(subject to constraint of given boundary values over dC_{M}) if y_{p}
satisfies the spacial Laplace equation Ñ_{p [e123]}^{2}y_{p} = 0
over C_{M} .
In electrodynamics we have L = - 2^{-4}p^{-1} f_{p}^{2}
+ c^{-1}j_{p}¿a_{p}
where c is scalar lightspeed and f_{p} = Ñ_{p}Ùa_{p} extremised by solutions to
Maxwell equations.
×××××××××××××××××××××××××××××××××××××××
References/Source Material for Multivector Physics
Anthony Lasenby
"Recent Applications of Conformal Geometric Algebra" 2005
http://www.mrao.cam.ac.uk/~clifford/publications
Leo Dorst. Daniel Fontijne, Steve Mann
"Geometric Algebra for Computer Science"
Morgan Kaufmann 2007
[Amazon US UK]
http://www.geometricalgebra.net
David Hestenes
"New Foundations For Classical Mechanics"
Kluwer Academic Publishers, 1999
Next : Multivector Relativity