The problem under consideration is a linear convection PDE.
\(\) From Farlow’s “Partial Differential Equations for Scientists and Engineers”, chapter 15 problem 3 is $$u_t = -2u_x \quad -\infty < x < \infty \quad 0<t<\infty $$$$ u(x,0)=e^{-x^2}$$
Intuition
We see that the governing equation is a linear convection problem. The characteristic velocity is 2. We expect the solution to be a shifted initial condition $$u(x,t)=e^{-(x-2t)^{2}}$$
This post provides a visual characterization of a generic flight envelope with a standard atmosphere. The following figure shows a generic flight envelope map.
Atmosphere and Airspeed
A pdf version is available at airspeed-2014c.pdf. This file plots altitude (0 to 50 thousand feet), calibrated airspeed (0 to 1000 KCAS), true airspeed, Mach number, dynamic pressure, static pressure, and total temperature on one handy page.
Airspeed:
Engineers and pilots track three different speeds:
Photo (c) Andy Wolfe
The F-35 Joint Strike Fighter (JSF) was conceived as a 5th generation fighter with a tri-service economy of scale. The late 1990s philosophy is stated as: “The rising unit costs of military aircraft and the new emphasis on greater commonality of aircraft designs among the services tended to push procurement trends toward ever smaller numbers of even more complex and expensive fighters designed to offer multirole and cross-service capabilities. [1]” A post-design analysis by RAND 20 years later estimates that the JSF’s life-cycle cost is 65% higher than a single-service fighter rather than the promised 16% lower life-cycle cost [2]. As of mid-2014, the F-35’s flight envelope is restricted to Mach 0.9, 3 g’s, and 3 hours between engine inspections [3]. Question: what is the subsequent impact to F-16 and A-10 mission replacements if the F-35 program continues having fielding issues?
[1] M. Lorell and H. Levaux, The Cutting Edge: A Half Century of U.S. Fighter Aircraft R&D, RAND, 1998.
[2] M. Lorell, M. Kennedy, R. S. Leonard, K. Munson, S. Abramzon, D. An and R. Guffey, Do Joint Fighter Programs Save Money?, RAND, 2013.
[3] A. Mehta, “Some F-35 Flight Restrictions Lifted,” DefenseNews, 2014. [Online]. Available: http://www.defensenews.com/article/20140729/DEFREG02/307290036/Some-F-35-Flight-Restrictions-Lifted.
Photo by TSGT CURT EDDINGS
The Dassault Mirage operational career spans 50 years in its many variants: mid 1950s until 2014. By modern standards, the aircraft design is exceptionally simple yet particularly capable.
[The Mirage III] was the first French aircraft to incorporate area rule technology and the first to exceed Mach 1.5 in flight. Nevertheless, Dassault employed only 14 engineers and draftsmen in its design and only 70 shop fabricators in its assembly…. The pre-production Mirage III A, which first flew in May 1958 reached a speed of Mach 2.2 and an altitude of 82,000 feet one month later. It differed from the earlier prototype chiefly in having an improved flight control system and somewhat more powerful Atar engine. [1]
The design time to first flight was an astonishing 9 months [1]. Dassault’s design groups operated on an incremental modification and prototype strategy. Designs failing to reach viability or exhibiting increasing risk were terminated cheaply and quickly (i.e. Mirage III G, variable sweep version). The Mirage was sold worldwide for many decades.
[1] R. Perry, A Prototype Strategy for Aircraft Development (RM-5597-1-PR), RAND, 1972.
What is a work of art? Can engineers create works of art? Or are creative works of art solely seen in performance arts?
This is the question under consideration in this post. Let us review some images.
http://www.bugatti100p.com/
Hughes H-1 Racer
Hughes H-1 Racer Tail
F-86
It seems that engineers don’t need to join a performance art troupe to create works of art.
Full memo: AEM495-2014-M02-revA
This post concerns a reverse engineered CAD model of the Cessna Citation 2. The objective of this memo is to provide a quick evaluation of the model with respect to the aerodynamics.
The model was supplied in a .STEP format by GrabCAD.com. It should be noted that the loft geometry presented is not an official Cessna geometry; the geometry was solely determined by reverse engineering. The as-supplied model required moderate loft-cleanup and surface trimming with CATIA. The cleaned model is illustrated in Figure 1. The forward hatch area was particularly difficult as the original model attempted to represent the hatch gaps with a tangential double fillet.
Figure 1: Citation CAD model (full aircraft)
A significant loft issue in the canopy area was spotted with zebra lighting (see Figure 2) at the intersection of the cylindrical fuselage and the complex canopy surface. From the zebra lighting, this region does not appear to be either tangentially or curvature constrained. This canopy loft issue will appear as a local accelerated flow region with detrimental downstream separation. A comparison with the actual Citation surface reveals that the current CAD model poorly represents reality.
Cp Contours: Mach 0.80, AOA 0
Aviation in itself is not inherently dangerous. But to an even greater degree than the sea, it is terribly unforgiving of any carelessness, incapacity or neglect.
—Captain A. G. Lamplugh
Dr. Sighard Hoerner’s Fluid Dynamic Drag and Fluid Dynamic Lift are classic references in aerospace engineering and fluid dynamics. These books are wonderful for rapid design and analysis work; I adore and treasure my copies. I have received multiple requests for information on purchasing.
As of 2022, the Hoerner family sells the Drag book at: http://hoernerfluiddynamics.com/. You can also reach them via email at: drag@hoernerfluiddynamics.com
As of 2024, the official source of these books is unknown.
The Lift book is unfortunately not available except on the used-market. But, you should still get it.
Please be aware that the books remain under copyright until 2045. These are worth the price to have your own copies.
The book details are:
Lift ISBN: 978-9998831636
Drag ISBN: 978-9991194448
Recently, Tuscaloosa received a sustained storm with significant rainfall. The video shows a hydraulic jump (shock) at Lake Tuscaloosa’s spillway (channel flow).
How close is my Fourier Sine series to a function f(x)?
Answer: \( L_2(N) = -\frac{1}{2} a_n^2 + \int_\Omega f^2(x) dx \)
This question came up recently while discussing PDE (partial differential equations) solution techniques. The final result is quite interesting.
Our sine series approximates the function f(x) over the domain \(\Omega\) as $$f(x) \approx \sum {{a_n}\sin (n \pi x)}$$ You can determine the coefficients with the formula. $$ a_n = 2\int_\Omega f(x) \sin(n \pi x) dx$$
One good measure of the error between the sine series and the function is the \(L_2\), pronounced “el squared”, error. $$ L_2 =\int_{\Omega} \left( u(x) – f(x) \right)^2 dx $$
Substitute for the sine series to obtain. $$ L_2 = \int_{\Omega} \left({{a_n}sin (n \pi x)}- f(x) \right)^2 dx $$
Expand the terms to obtain $$ L_2 = \int_{\Omega} a_n^2 \sin^2(n \pi x) dx -2\int_{\Omega} a_n f(x) \sin(n \pi x) dx + \int_{\Omega} f^2(x) dx $$
Now, these integrals are particularly interesting. The first integral is a constant \(0.5 a_n^2\). The second contains the definition of \(a_n\). The third only contains the function \(f^2(x)\). This simplifies to $$L_2(N) = -\frac{1}{2} a_n^2 +\int_\Omega f^2(x) dx$$
Interestingly, if we let the error become zero, the following is an identity $$ a_n^2 = 2 \int_\Omega f^2(x) dx$$ Knowledge of this identity will allow you to quickly compute integrals of squared trig functions.
Neat! Do you have any interesting Fourier results? Let me know in the comments.