Model fit Supersonic

13e20851-00c2-4dc2-a06e-c3cfe43b6ea9 37 4 5 decimal

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Model Supersonic

k 0.88128485 :

x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases :

XY x 1153 range : j 1 x rows range sup j el x j el Supersonic : for x sup 6 round vectorize augment 31 line :

1 0.25 0.132 1015 mat transpose

k 112 mat XY stack iter13 12 mat

XY XY eval :

k β f 1 - 0.001 - ≡

Minimize Conjugate-Gradient

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

init iter Minimize XY f φ β LS X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx-> raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line : β 1 el init : i 1 : i iter ≤ β i 1 + el XY f φ β i el LS : i i 1 + : 21 line while β β iter el : 41 line β 31 line :

Vector Gradient ... Optimiz Symbolic
seemingly a scalar kernel  not 
recognized in program loop x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

corr Pearson correlation χ X Mean : μ Y Mean : n XY rows : 13 mat X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el β f eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el β f eval μ - (2^(i 1 n sum * sqrt / / 31 line :

correlation between [X,Y data]

correlation ... [X,fitted model]

Absolute  residuals observer
fetches β from calculated δ j 1 XY rows range Δ j el Y j el X j el β f - eval : for X Δ augment 21 line :

SSD = "Sum Square Differences" SSD Y i el X i el β f - (2^(i 1 XY rows sum :

data 1 1.0466 3 1.6474 5 2.069 7 2.4167 9 2.7198 11 2.9921 13 3.2415 15 3.473 17 3.69 19 3.8949 21 4.0895 112 mat :

Model function x β f β 1 el x β 2 el^* β 3 el x ln * + β 4 el x sqrt * + β 5 el + :

<= ConjugateGradient Vector Fnct x β φ x β f 5 φ :

X XY 1 col : Y XY 2 col : 12 mat data data . 12 black plot : n XY rows : 31 line

1 0.3 01051 mat

init 1 0.3 01015 mat transpose :

iter 7 :

algo solving for β β init iter Minimize :

assign/isolate .. β β :

SSD 1107 -^*

corr 0.999996

vx data 1 col : vy data 2 col : 12 mat

β 0.008944 - 7.047011 - 0.059936 - 0.897978 0.15752 51 mat

Relative residuals δ i 1 data rows range Δ i el vy i el vx i el β f - eval : for vx Δ augment 21 line :

x λ 1 k x ≤ cases :

data x β f x λ * 2 1 sys δ 0.125 BarSize

δ 103 0.0004 50709011013015 0.0001 - 17 0.0002 - 19 0.0003 - 21 0.0006 - 112 mat

k β f 1 - 0.0137 -

k 0.88128485 :

x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases :

data x 1213 range : j 1 x rows range sup j el x j el Supersonic : for x sup 6 round vectorize augment 31 line :

data data eval : 1 1.0466 3 1.6474 5 2.069 7 2.4167 9 2.7198 11 2.9921 13 3.2415 15 3.473 17 3.69 19 3.8949 21 4.0895 112 mat

If the speed of sound is not known, Mach number may be determined by measuring the various air pressures (static and dynamic) and using the following formula that is derived from Bernoulli's equation for Mach numbers less than 1.0. Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is:[8]. The formula to compute Mach number in a supersonic compressible flow can be found from the Rayleigh supersonic pitot equation (above) using parameters for air: where: qc is the dynamic pressure measured behind a normal shock.

The all thing is solving for 'M' that appears on both sides. Hre comes the power of Smath solving as given, solving on the canvas.

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

t0 1 time :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 20 black 1 5 mat 3 1 sys

1 time t0 - 69 s *

supersonic

subsonic

Mach 1.0 is the speed of sound in air, so a plane flying Mach 2.0 is flying twice as fast as the speed of sound. The speed of sound is not a constant, but depends on altitude (or actually the temperature at that altitude). A plane flying Mach 1.0 at sea level is flying about 1225 km/h (661 Knots, 761 mph), a plane flying Mach 1.0 at 30000 ft is flying 1091 km/h (589 knots, 678 mph) etc. Speeds below Mach 1 are called subsonic, between Mach 0.8-1.2 Transonic and above Mach 1.2 Supersonic.

Source data plotG(vx,vy,char,size,color)

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Mach ► km/hr ... at cockpit altitude(m) x Cf 1225 x 70.884 - x 3048 - 2826.533 x 6006 - 3.553 - / + / + / + :

data 01225152412043048118245721160600611397620111591441090106681063121921062137161062152401062167641062182881062132 mat :

vx data 1 col : vy data 2 col : 12 mat source vx vy . 10 black plotG : 21 line

x λ x Cf 0 x ≤ (x 10668 ≤ (& 106210668 x ≤ (x 20000 ≤ (& cases :

Mach correction vs cockpit altitude(m)...

source x λ 2 1 sys

Mach 1 :

Mach 8500 λ * km hr / * Mach 8500 λ *'km'hr / * 21 mat 306 m s / * 1101'km'hr / * 21 mat

Mach 2.5 :

Mach 5000 λ * km hr / * Mach 5000 λ *'km'hr / * 21 mat 801 m s / * 2885'km'hr / * 21 mat

on-fly dual result Mach 1.5 : altitude 10000 : 12 mat

make the unit system 'silent'
so to explicit the dual result Mach altitude λ * km hr / * Mach altitude λ *'km'hr / * 21 mat 448 m s / * 1613'km'hr / * 21 mat

Thiele transaction to J_Frac J(x)

x f 1225 x 70.884 - x 3048 - 2826.533 x 6006 - 3.553 - / + / + / + :

Step: 1  ... Assign Symbolic ... Optimiz = NONE x collect x f confrac x convert maple :

x collect 1000 x * 74437 / - 621250311249927503715179000 / + 109601814613830134830327353852160154073250000000 x 261005361876391691750000 / - (* / +

x J 0.013434 x * - 1233.3365 + 1.2862 105^* x 15428.1284 - / + :

x f 0 x ≤ ( x 10668 ≤ ( & * x J 10668 1063 + 20 black 1 5 mat 3 1 sys

Mach Number

The Cauchy parameter, after transformation becomes the 'Mach number' subsonic for 'Air' as ideal gas is derived from Bernoulli equation.

M 2 γ 1 - / (qc p / 1 + (γ 1 - γ /^1 - (* sqrt ≡

In supersonic condition, compressible fluids and past the "choc wave", Mach number is derived from the Rayleigh-Pitot equation with 'qc' = impact pressure past the "choc wave".

M 0.88128485 qc P / 1 + (117 M 2^* / - (2.5^* sqrt * ≡

The static ports of Pitot tube are always multiple. The principle behind is ChebyShev kind of averaging. A similar technique applies to "Annubar" a flow measuring device for dirty fluids.

Recover Pitot 'x'
from given Mach. x U Pitot U x 1 + (117 U 2^* / - (2.5^* sqrt / k - : (0 ≡ U GivenMach x U Pitot x solve maple : U Pitot (k 2^17 U 2^* 3 - 7 U 2^* 37 U 2^* - (* + (* + (* 49 U 8^* 1 - 7 U 2^* + U 2^/ sqrt * 7 sqrt * + 1 - 7 U 2^* + (k 2^* 17 U 2^* 2 - 7 U 2^* + (* + (* / : 31 sys

Given Mach... Find Pitot ratio 'x'
sanity recover Pitot 'x' from Mach
random Mach from Pitot ratio 'x'
Mach ... Bombardier Global 7000 5 Pitot (5 Pitot Supersonic 1.75 Supersonic 0.788 Subsonic 41 mat 31.653 5 1.311 0.95 41 mat

NO explicit supersonic Mach(x) was found

1. only numerical solver roots

2. alternately: dedicated Mach(x,β)

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for convenience Pitot 'x' [0.75 ... 50] x mach 7.5 x * 170529 x 0.87 - (0.528^* + 2495 - 14774 x 0.528^* + / * abs :

k mach 0.997

Supersonic β 21.7077 233.903 20.0537 4.31361 0.165479 4276.76 491.007 16.7228 0.559674 91 mat :

5 Supersonic 35 Supersonic 40 Supersonic 31 mat 2.069 5.254 5.611 31 mat

Supersonic Mach segment [J_Fraction]
some "Autorithy" decided  Mach > 5 hypersonic x β Mach β 1 el β 6 el x β 2 el + β 7 el x β 3 el + β 8 el x β 4 el + β 9 el x β 5 el + / - / - / - / - :

5 β Mach 35 β Mach 40 β Mach 5 mach 25 mach 40 mach 61 mat 2.069 5.253 5.610 2.066 4.453 5.655 61 mat

β 0.0089 - 7.047 - 0.0599 - 0.898 0.1575 51 mat :

Model function x β f β 1 el x β 2 el^* β 3 el x ln * + β 4 el x sqrt * + β 5 el + :

5 β f 35 β f 40 β f 31 mat 2.069 5.257 5.616 31 mat

2.069 5.254 5.611 31 mat

Solving Pitot ratio 'x' given Mach(U)

Pitot_sub(U) . . . Pitot_sup(U)

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U Pitot_sub 1 - 2 U 2^γ * + U 2^- 2 / ln γ * 1 - γ + / exp + :

x sub 10 x ≤ (x 1 ≤ (& cases : x sup 11 x ≤ (x 2 ≤ (& cases : 21 sys

Explicit Pitot ratio 'x' 
from given Mach = U U Pitot_sup k 2^17 U 2^* 3 - 7 U 2^* 37 U 2^* - (* + (* + (* 49 U 8^* 1 - 7 U 2^* + U 2^/ sqrt * 7 sqrt * + 1 - 7 U 2^* + (k 2^* 17 U 2^* 2 - 7 U 2^* + (* + (* / :

subsonic
hyersonic 0.75 Pitot_sub 5 Pitot_sup 21 sys 0.452 31.653 21 sys

unknown interest: Find Pitot 'x' <= Given Mach 'U'

x Pitot_sub x sub * x Pitot_sup x sup * 1 k + 20 black 1 5 mat 3 1 sys

These two inverse functions 1. Pitot_sub(U) 2. Pitot_sup(U) given for unknown interest, possibly testing the Pitot in-situ installation.

Refreshed:20220426

Mathcad scalar plot from assigning Find(M) to a wild function