AE 301 Aerodynamics I
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Transcript AE 301 Aerodynamics I
Compressible Potential Equation
•
Although we have seemed to have strayed – remember
that we are still solving a PDE. In particular:
2
2
2
2
2 2 2 0
x
y
z
•
This potential equation was derived under the
assumption of incompressible, irrotational flow.
•
Let’s know revisit the original equations and consider
how this relation would be differ with compressibility.
•
We will also see how to apply the potential flow concept
to high speed (transonic and supersonic) flows.
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Technically, flow with shock waves is rotational, but only
slightly if the shocks are weak.
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Compressible Potential Equation [2]
•
Our starting point is the Euler conservation equations
we derived at the start of the semester.
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If we assume steady flow, mass and momentum
conservation can be written as:
V 0
V V p 0
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In incompressible flow, the first equation was enough to
derive our potential flow relation.
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This time, we will concentrate our attention primarily on
the second equation, momentum conservation.
•
We will also need the energy equation, but let’s
consider that a little later.
AE 401 Advanced Aerodynamics
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Compressible Potential Equation [3]
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We can “combine” mass and momentum conservation
through chain rule expansion of momentum:
V V V V p 0
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The first term, by continuity, vanishes. The remaining
terms, expanded in 2-D become:
u
•
u
u
p
v
x
y
x
v
v
p
v
x
y
y
If the potential function exists, then:
V
•
u
ˆ ˆ ˆ
i
j
k
x
y
y
Note, we have switched to using capital phi for our
symbol – the reason will be apparent later.
AE 401 Advanced Aerodynamics
200
7/18/2015
Compressible Potential Equation [4]
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The definition of the potential function gives:
u
•
x
•
y
With these, momentum conservation becomes:
2
2
p
x x 2
y xy
x
•
v
2
2
p
x xy
y y 2
y
Since these equations are getting long, let’s introduce a
common notational simplification – indicate the partial
differentiation through subscripts:
p
x xx y xy
x
x xy y yy
p
y
Next, since we are compressible, we know we should
introduce some thermodynamics. Let’s do that with the
definition of the speed of sound.
AE 401 Advanced Aerodynamics
201
7/18/2015
Compressible Potential Equation [5]
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The speed of sound is:
a2
•
•
p
If we allow ourselves to be a little loose with our
calculus, this can be written as:
dp
d 2
a
Or, since density and pressure are functions of location:
1 p
y xy
x a x
a
1 p
2
2 x xy y yy
y a y
a
•
2
2
x
xx
To finish up, return and consider the continuity again.
AE 401 Advanced Aerodynamics
202
7/18/2015
Compressible Potential Equation [6]
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If we expand the continuity equation:
V V V
u v
u
v
0
x
y
x y
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Introducing the potential function gives:
x y
xx yy
0
x y
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Finally, inserting our previous results from the
momentum/speed of sound equation:
y
x
xx yy 2 x xx y xy 2 x xy y yy 0
a
a
AE 401 Advanced Aerodynamics
203
7/18/2015
Compressible Potential Equation [7]
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Or, after combining like 2nd order differentials:
2y
2 x y
2x
1 2 xx 1 2 yy
xy 0
2
a
a
a
•
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It is fairly easy to see that as the Mach number goes to
zero, this equation reduced to the Laplace equation:
2
2x y u 2 v 2
2
M
a2 a2
a2
However, this new equation, unfortunately, is a highly
non-linear PDE with no easy solutions.
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This equation is also not yet complete – the speed of
sound terms are not constants but vary with speed.
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Thus we need the energy equation.
AE 401 Advanced Aerodynamics
204
7/18/2015
Compressible Potential Equation [8]
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Rather than use the differential form of the energy
equation, lets use the algebraic equation from Aero 2:
V2
c pT
c pT0
2
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Remember those cp’s are specific heats, NOT pressure
coefficients!
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This can be using our equations for cp and the a:
RT V 2 RT0
1 2 1
a02
a2 V 2
1 2 1
a a
2
2
0
AE 401 Advanced Aerodynamics
1
2
V a
2
2
0
1
2
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2
x
2y
7/18/2015
Compressible Potential Equation [9]
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To solve these equations, we could use a numerical or
Computational Fluid Dynamics (CFD) approach:
– Express the 2nd derivatives terms using some numerical method
like Finite Differences, Finite Volumes, or Finite Elements.
– Asssume values of velocity (1st derivatives) and a, solve for .
– Update the velocities and a and start iterating.
– (Easier said then done, believe me!)
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Or, we can look at approximate solutions obtained by
“linearizing” the equations.
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This process involves expanding the terms we have
using a perturbation potential, and then throwing away
terms of lower magnitude.
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That is where we are going next.
AE 401 Advanced Aerodynamics
206
7/18/2015