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Homework 9: Shock / Expansion Theory
Handed out Due Note: This assignment is worth 2 homework grades 1. For a given Prandtl—Meyer expansion, the upstream Mach number is 3 and the pressure
ratio across the wave is 0.4. Calculate the angles of the forward and rearward Mach lines of the expansion fan. relative to the free—stream direction. 2. Consider a supersonic: ﬂow with an upstream Mach number of 4 and a pressure of 1
atm. This flow is ﬁrst expanded around a corner with 8 x 15 deg, and is then compressed
through a corner with an equal angle so that it is returned to its original upstream direction.
Calculate the Mach number and pressure downstream of the compression corner. 3. A supersonic get exits from a nozzle with an exit—to—throat area ratio of 3.0 into a region
Where the back pressure is 0.?98 X 105 Pa. The reservoir pressure is 10 X 105 Pa, and the
nozzle is overwexpanded. Given the sketch shown below, find the angles of the slip line :9 and 6 and the Mach numbers and pressures in regions (2M3)1 and nil W. PE) 3»; E. W a 4. (Sampie Test Problem) Mach 4 air ﬂows over the airfoil shown below. a. Sketch and label the wave patterns present in this ﬂow h. Determine expressions for the lift and drag coefﬁcients in terms of the pressure
coefﬁcients in regions 1,2, and 3 and the geometry. c. Calculate the pressure coefﬁcients in each region, using shockexpansion theory.
d. Using your resuits from and (o), calculate the lift and drag coefﬁcients.
9 O
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Heme—Lt ~———e—— ( I “1 /
’1‘: 5. (Sample Test Problem) The ﬂow in the exit plane of a GE) nozzle merges with another 1 ﬂow outside the nozzle. The two ﬂowfields are separated by a slip Zine inclined 10 deg.
upward, as shown. The nozzle has a reservoir pressure of 5 atm and an exit—to—throat area
ratio of 25. The Mach number of the outside ﬂowfield is 3.0 upstream of the lip of the nozzle.
If the flow in the divergent section of the nozzle is supersonic and isentropic. a. Sketch the wave interactions that wiil occur in this ﬂow, inciuding those downstream
of point A. b. Caicuiate the Mach number and pressure at the exit plane of the nozzle. c. Calculate the pressures on both sides of the shp line (before point A) and aiso the
pressure of the ﬂow outside and upstream of the nozzle lip. M:2.CD A
nun/Q10.
1’ m—s‘v
ha 5. (Computer Problem) Consider Mach 3 ﬂow past a double wedge geometry, as discussed
in class. Modify your computer codes to calculate the angle that the slip line makes With
the X axis and the pressures in Regions 4 and 5. Consider two cases, with each having 91 =
12 degrees and 92 m 15 degrees, but with the ﬁrst case assuming that the coalesced shock is
a weak obiique shock and the second assuming that it is a strong oblique shock. You may
wish to verify whether the additional wave generated is a compression wave or an expansion
wave before proceeding. Use the attached pieces of code to get a Mach number, given the
PrandtlMeyer function, for the case where an expansion fan is present. Turn in a copy of
your code as weh as output tabulating your Merit function versus number of iterations for both cases.
£95651 5 37mph at
/ Co [\3 function rpm(xx)
mm____%m____W____mMmmMwﬂ____________MW__WWWW____~___
this function inverts the PMM function to get the Mach number. A ﬁinitemdifference Newton
method is used. This routine is accessed from the main program'
by a statement such as xmach = rpm(xnu) where_xnu is the prandtl—meyer function (in RADIANS)
and xmach is the returned Mach number W 00000000000000?) rpm 2 1.05
dxm = 0.00; do i=1,35
fx = pm(rpm)—XX
if(i .eq. 1) fXO : fx
if(abs(fx/fx0) .le. le~6) goto 998
dfxdx : (pm(rpm+dxm) — pm(rpm))/dxm
rpm 2 rpm _ fx/dfxdx
enddo
write(6,*) ’convergence not reached“ ‘998 continue
return
end function pm(xx)
__wmmmm___n__mWW___ia______mw____________~._______~.____mmﬁ
this function computes the Prandtl—Meyer function given the Mach number XX. The P—M function is in
RADIANS ——.~wmwm___ﬂmwm_~___mmmmmmwm—___M___ﬂ_ﬂ—__H_ OODOOO pm 2 sqrt(2.4/O4)*atan(5qrt(0.4*(xx*xx—l.)/2.4))
c — atan(sqrt(XX*XX—1.M return
end ...
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 Spring '11
 Dr.Edwards
 Aerodynamics, Mach number, Shock wave

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