Questions and Answers of Fundamentals Of Aerodynamics

Define the critical Mach number for subsonic flows. Describe how it is determined for an airfoil.
Plot the lift line slope change of a thin wing with respect to the aspect ratio.
Plot the lift line of a swept wing with a low aspect ratio using Diederich formula with respect to sweep angle for $\mathrm{AR}=2,3,4$.
Find the wave drag of an $8 \%$ thick biconvex airfoil at free stream Mach number of $\mathrm{M}=2$.
For a thin airfoil pitching simple harmonically about its leading edge, plot the amplitude and phase curves with respect to the reduced frequency at transonic regime.
Compare the amplitude of a sectional lift coefficient of a thin airfoil in vertical oscillation in transonic regime with the same airfoil oscillating in incompressible flow in terms of the reduced
By definition, if the change of the moment about the center of gravity of a slender body with respect to angle of attack is negative then the body is statically stable, Fig. 1.7. Comment on the
Compare the hypersonic surface pressure expression with the incompressible potential flow surface pressure of a flow past a circular cylinder. Comment on the validity of both surface pressures.
Find the surface pressure for the frontal region of the capsule during its reentry. Assume the shape of the frontal region as a half circle and comment on the region of validity of your result.
Find the amplitude of the surface pressure coefficient for a flat plate simple harmonically oscillating in hypersonic flow with amplitude $\bar{h}$. Define an interval for the hypersonic similarity
For a delta wing with a sharp leading edge separation plot the non dimensional potential $K_{p}$ and vortex lift coefficient $K_{v}$ changes with respect to the aspect ratio $\mathrm{AR}$.
Explain why we need to resort to unsteady aerodynamic concepts for ornithopter studies.
In a barotropic flow show that $\frac{1}{ho} abla \mathrm{p}=abla \int \frac{d p}{ho}$.
Eq. 2.15 is written in terms of the velocity potential. Express the same equation with partial derivatives of velocity potential.Eq. 2.15 D A - 1 (8 a 2) 5 2 q + t .b+ = 0
An oblate ellipsoid is undergoing vertical simple harmonic motion with amplitude $\bar{a}$. Express the equation of upper and lower surfaces of the airfoil.
Comment on the steady or unsteady lift generation by referring to the downwash expression given by 2.19.Eq 2.19 W = +u +v
The equation of a paraboloid of length 1 and whose axis is in line with $\mathrm{x}$ axis is given as \(c(\mathrm{x} / \mathrm{l})=\left(\mathrm{y}^{2}+\mathrm{z}^{2}\right) / a^{2}, 0 \leq x \leq
A lighter than air prolate ellipsoid moves in air with constant speed $U$. If this air vehicle oscillates simple harmonically about its center with a small amplitude $\mathrm{A}$ in a vertical
We do not need to define perturbation potential for the acceleration potential. Why?
From the non linear relation between the velocity and the acceleration potential, obtain the linear relation given by Eq. 2.25.Eq. 2.25 = +U.
Obtain the surface pressure and downwash expressions in terms of acceleration potential.
Obtain Equation 2.35 which gives the continuity of the speciesEq 2.35 DCi Dt P =V.(pDmici)+Wi
Express the conservation of momentum in open form in Cartesian coordinates.
Obtain the expression given by 2.50 by means of the transformation from Cartesian to generalized coordinates.Eq 2.50 JF OF OF + +G + ' =
For a tapered wing with half-span of 4 units let $\mathrm{x}$ be the chordwise and $\mathrm{y}$ be the spanwise directions. The equation for the leading edge is given as: \(\mathrm{x}=0.15
In generalized coordinates, obtain the Navier-Stokes equations for the thin shear layercase in terms of the contravariant velocity components.
Equation 3.3 gives the relation between the downwash and the vortex sheet strength in $\mathrm{x}-\mathrm{Z}$ coordinates for a positive free stream running from left to write. Obtain a similar
An airfoil is given by a parabolic camber line, i.e., $z_{a}=-\left(a / b^{2}\right) x^{2}$. Find: (i) sectional lift coefficient, (ii) center of pressure, (iii) aerodynamic center, at zero
Find the phase difference between the displacement and the downwash for a flat plate oscillating simple harmonically in a free stream at a zero angle of attack.
Comment on the physical meaning of the Theodorsen function.
Find the sectional moment coefficient taken about the midchord of the Example 3.4. Plot the change with respect to angle of attack.Example 3.4Find the sectional lift coefficient change for an airfoil
Find the lift and moment coefficients about the quarter chord for NACA 0012 profile which is pitching about its quarter chord with $\alpha(t)=3^{\circ}+10^{\circ} \sin \omega t$. (Compare your
For the returning wake problem, interpret the phase angles for, $\omega / \Omega$ values being equal to an integer, integer plus a quarter and integer plus one half. Take $h / b=3$. Find the
Use the data of Problem 3.8 to find the phase differences of the Loewy function for a double bladed rotor where only the distance between the blades are counted. Make the same computations for
Obtain the time variation of the sectional lift coefficient for an airfoil which is pitching about its leading edge as shown in Fig. 3.15 using (i) unsteady aerodynamics, (ii) quasi unsteady
Find the lift and the moment coefficients about the midchord of an airfoil which undergoes a sudden vertical translation under zero angle of attack. Use the Wagner function.
Find the lift and the moment coefficient at the midchord of an airfoil undergoing sudden velocity change. Use the Wagner function.
If the gust intensity with time varies as given in Fig. 3.16, obtain the lift and the moment coefficient changes about the midchord and their plots with respect to time.Fig. 3.16 Wg 0 to t
Express the phase angle of the Sears function as the function of the reduced frequency and compare it with the phase of the Theodorsen function on a graph.
Consider the simple harmonically varying free stream problem for the reduced values $k=0.2,0.4,0.6$ and $0.8$, find the lift coefficient under constant angle of attack. Take the amplitude
For a simple harmonically varying free stream problem obtain the expression for sectional moment coefficient about the midchord in terms of the reduced frequency for the amplitude. Plot the graph for
Assume a blade with radius $R$ is rotating with angular speed $\Omega$ at a constant forward flight speed $U$. Show that the problem can be modeled as a variable free stream: \(U s=U \sin
The vorticity vector, $\boldsymbol{\omega}$, is defined from the velocity vector, $\mathbf{q}$, as follows $\boldsymbol{\omega}=$ $abla \mathbf{x} \mathbf{q}$. Show that the vorticity vector
Evaluate integral $I_{1}$ of the downwash expression 4.9 with integration by parts.Eq 4.9 Wa(x, y) == ====7 // R - =- Ra - A JJ R [(x ) + (y 1)/2dd Va(, n)(x ) + da(, n) (y n) dan
Transform the spanwise $\mathrm{y}, \eta$ coordinates into $\phi, \theta$, and obtain Eq. 4.21 as Glauert did.Eq. 4.21 ab = aobo An sin no+ n=1 bn sin no 21 sin
The Aerodynamic Influence Coefficient Matrix, [A], gives the lift coefficient generated at a section with the angle of attack change at another section, i.e., $\left\{c_{l}\right\}=[A]\{\alpha\}$.
Find the Aerodynamic Influence coefficients for the wing given in Fig. 4.7.Fig. 4.7 a.c 2.8m 7m 1.4m
Show that for an elliptically loaded wing, the total lift line slope in terms of the root lift line slope $a_{o}$ and the aspect ratio AR reads as\[\frac{d c_{l}}{d \alpha}=a_{o} \frac{A R}{A R+2}\]
Glauert's, G(ϕ)=Γ(ϕ)/(2lU)=∑mi=12AisiniϕG(ϕ)=Γ(ϕ)/(2lU)=∑i=1m2Aisin⁡iϕ, series can be written with Multhopp's distribution for ϕϕ as ϕn=nπm+1,n=1,2,…,mϕn=nπm+1,n=1,2,…,m. , to
In Weissinger's L-Method the circulation G can be expanded into Fourier series after performing the $y *=\cos \phi_{j}$ transformation. Using the trapezoidal rule for integration the angle of
Find the center of pressure for a thin wing with small aspect ratio.
If there is a chordwise camber for the wing given in Example 2 find the lift coefficient at zero angle of attack. Does parabolically cambered airfoil satisfy the Kutta condition? How do we have to
If there is a spanwise parabolic camber in Example 2 find the lifting pressure coefficient for the wing.Example 2 For a low aspect ratio delta wing with angle of attack $\alpha$, plot the
Obtain the time dependent lift coefficient for an elliptic thin wing in a variable free stream. For a wing with an aspect ratio of 6 obtain the unsteady to quasi steady lift ratio for the Example
The tapered symmetric thin wing geometry is shown in Fig. 4.8. If this wing under goes a simple harmonic motion in vertical direction, find the amplitude of sectional lift coefficients in terms of
Find the total lift coefficient for the wing given in Example 4 withi) theoretical approach, ii) with empirical correction.Example 4The wing given in Example 2 is undergoing a simple harmonic motion
For a delta wing pitching simple harmonically about its nose, with respect to pitch angle find the amplitudes of,(i) lifting pressure coefficient,(ii) chordwise and spanwise lift loadings,(iii) total
For a delta wing rolling simple harmonically about its root find the amplitudes of(i) lifting pressure,(ii) spanwise and chordwise variation of lift,(iii) total lift coefficient, with respect to
Show that Eq. 5.1 given for incompressible source satisfies the Laplace's Equation.Eq. 5.1 (x, y, z) = q 2 4x + y+z
With Gallilean type transformation show that Equation 5.3 transforms into Laplace's Equation.Eq. 5.3 x2 + y2 12 = 0
Show that $\bar{g} \frac{e^{ \pm i k \bar{R}}}{\bar{R}}$ is a solution for the Helmholtz Equation.
Show that the perturbation potential equation expressed in moving coordinates for a steady compressible flow, satisfies Eq. 2.24 in original coordinates.Eq. 2.24 V' 1 +U- $' = 0 t x
Find the solution given by Eq. 5.7 for the relation between the time, $\tau$, of initiation of the disturbance and the present time $t$, in terms of Mach number.Eq. 5.7 +1=2 1 5 [Mx x + B (y + z
Show that Eq. 5.10 given for the subsonic unsteady potential satisfies the differential equation given by Eq. 2.24.Eq. 5.10Eq. 2.24 (x, y, z, t) = qe af i{t+Mx-x+B (x + z)]} 4x+(y+z)
Obtain the subsonic doublet expression Eq. 5.25 with taking the derivative of Eq. 5.24 with respect to $\mathrm{z}$.Eq. 5.25Eq. 5.24 (x, y, z, t) = -Ae+(Mx-R)] t)-Ae" (Mx-R) R iw ( + a R
Obtain the relation Eq. 5.26 between the pressure discontinuity and the doublet strength using dimensional analysis.Eq. 5.26 A = . - Poo
Using the general solution for the first order differential equations obtain the relation between the velocity potential and the acceleration potential.
The choice of $\cot (\theta / 2)$ for the chordwise variation of the pressure coefficient gives the $1 / \varepsilon^{1 / 2}$ type singularity at the leading edge and $\varepsilon^{1 / 2}$ type
Obtain the amplitude distribution for the lifting pressure of an airfoil plunging simple harmonically with amplitude $\bar{h}$.
Comment on the term $\sqrt{1-\eta^{2}}$ written as the coefficient of the polynomial series expressed for the spanwise direction.
A thin wing with aspect ratio 3 is oscillating simple harmonically in bending mode with the following amplitude distribution along span\[ \bar{h}(y / l))=0.18043(y / l)+1.70255(y / l)^{2}-1.13688(y /
The wing-tail interaction problem for $\mathrm{k}=0.75$ and $\mathrm{M}=0$ is shown in Fig. P5.15.For the simple harmonic plunging obtain the lift amplitude. Take 10 panels in spanwise direction
A wing with small aspect ratio and a store at the tip is shown in Fig. P5.16. For the simple harmonic plunging of the wing at $\mathrm{k}=0.86$ and $\mathrm{M}=0$ find the spanwise lift
For Mach number 0.6, obtain the lift response of the airfoil to the arbitrary motion given by Problem 3.10.Problem 3.10Obtain the time variation of the sectional lift coefficient for an airfoil which
For Mach number 0.7, find the lift response of the airfoil subjected to the gust given by Problem 3.13.Problem 3.13If the gust intensity with time varies as given in Fig. 3.16, obtain the lift and
Using the Lorentz transformation given for the supersonic flow obtain the classical wave equation for the linearized potential.
Show that the solution for the classical wave equation satisfied by the supersonic potential is given by Eq. 5.28.Eq. 5.28. (x, y, z, t)= (, ) = {e' Poo z it+M(x)/R}ddn, - R = (x 5 ) + B [(y
Find the solution to the quadratic Eq. 5.59 a,b to obtain two different times for the initiation of the disturbances, 5.59 a,b.Eq. 5.59 (a,b) a(t 1) = x - U(t - t)] + y+z a(1-2)= U(12) x] + y+z - -
For the supersonic flow, show that the area of the integration over the surface is hyperbola. The chordwise integration process has the upper surface denoted by $-\mathrm{z}$, and the lower surface
Using the Euler's formula which gives the relation between the exponential function and the trigonometric functions, obtain the inner integral for the velocity potential expression given by 5.76.Eq
Find the lifting pressure amplitude for a thin airfoil pitching about its leading edge with $\alpha=\bar{\alpha} e^{i \omega t}$ in $\mathrm{M}=1.5$. Obtain the lift amplitude also.
A delta wing with sweep angle of $45^{\circ}$ undergoes a simple harmonic plunge oscillations at $\mathrm{M}=1.054$ at $\mathrm{k}=0.525$. Find and plot the real part of the velocity
A $45^{\circ}$ sweep angle delta wing is in the supersonic flow with $\mathrm{M}=1.5$.(i) Divide the half of the wing with equally sized 25 boxes and give identification numbers for the boxes
Show that for a steady supersonic flow the lifting pressure coefficient becomes\[ C_{p a}(x, y)=-\frac{4}{\pi} \frac{\partial}{\partial x} \iint_{V} \frac{w(\xi, \eta)}{U} \frac{d \xi d \eta}{R} .
Find the spanwise variation of lift for a delta wing with sweep angle $60^{\circ}$ at $\mathrm{M}=1.5$ and $\mathrm{k}=0.2$, at simple harmonic plunge. (Use 50 Max box for a half wing).
Obtain and plot the lift response for the arbitrary motion of the airfoil given in Problem 3.10 at $\mathrm{M}=2$.Problem 3.10Obtain the time variation of the sectional lift coefficient for an
Find and plot the lift response of the airfoil at Mach number of $\sqrt{2}$ experiencing the gust described in Problem 3.13.Problem 3.13If the gust intensity with time varies as given in Fig. 3.16,
For a supersonic flow under which condition the root pressure distribution of a thin wing can be determined with 2-D analysis. Why?
The surface equation for a slender body shown in Fig. P5.33 is given by\[ z=0.05 \sqrt{x / l}, 0 \leq x \leq l . \]Find the following derivatives of the aerodynamic coefficients,\[ \text { i) }
Starting with Eq. (7.9), derive Eqs. (7.10) and (7.11).\[\begin{align*}& e_{2}-e_{1}=\frac{p_{1}+p_{2}}{2}\left(\frac{1}{ho_{1}}-\frac{1}{ho_{2}}\right) \\&
Consider a normal shock wave moving with a velocity of $680 \mathrm{~m} / \mathrm{s}$ into still air at standard atmospheric conditions ( $p_{1}=1 \mathrm{~atm}$ and $T_{1}=288 \mathrm{~K}$
For the conditions of Prob. 7.2, calculate the total pressure and temperature of the gas behind the moving shock wave.Data From Problem 7.2:Consider a normal shock wave moving with a velocity of
Consider motionless air with $p_{1}=0.1 \mathrm{~atm}$ and $T_{1}=300 \mathrm{~K}$ in a constant-area tube. We wish to accelerate this gas to Mach 1.5 by sending a normal shock wave through the
Consider an incident normal shock wave that reflects from the endwall of a shock tube. The air in the driven section of the shock tube (ahead of the incident wave) is at $p_{1}=0.01 \mathrm{~atm}$
The reflected shock wave associated with a given incident shock can be calculated strictly from the use of Table A.2, without using Eq. (7.23). However, the use of Table A. 2 for this case requires a
Consider a blunt-nosed aerodynamic model mounted inside the driven section of a shock tube. The axis of the model is aligned parallel to the axis of the shock tube, and the nose of the model faces
Consider a centered, one-dimensional, unsteady expansion wave propagating into quiescent air with $p_{4}=10 \mathrm{~atm}$ and $T_{4}=2500 \mathrm{~K}$. The strength of the wave is given by
The driver section of a shock tube contains He at $p_{4}=8 \mathrm{~atm}$ and $T_{4}=300 \mathrm{~K}$. $\gamma_{4}=1.67$. Calculate the maximum strength of the expansion wave formed after
The driver and driven gases of a pressure-driven shock tube are both air at $300 \mathrm{~K}$. If the diaphragm pressure ratio is $p_{4} / p_{1}=5$, calculate:a. Strength of the incident shock
For the shock tube in Prob. 7.10, the lengths of the driver and driven sections are 3 and $9 \mathrm{~m}$, respectively. On graph paper, plot the wave diagram ( $x t$ diagram) showing the wave

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