Questions and Answers of Fundamentals Of Aerodynamics

Using Maslen method, find the approximate value of pressure and density at the junction of the sphere and the cone of Problem 7.29 at Mach number 8.Problem 7.29An empirical way to determine shock
Solve Example 7.5 with equilibrium energy as the initial condition for the Oxygen molecule at 1 atmosphere and \(3200 \mathrm{~K}\).Example 7.5Obtain the time dependent expression for the vibration
If the waverider of the Problem 7.44 has the wall temperature of \(1400 \mathrm{~K}\) and \(\operatorname{Re}=1.371 \times 10^{6}\) with respect to its length then find the drag coefficient with
What is the effect of flow separation at (i) swept,(ii) unswept wings at high angle of attack.
Obtain an explicit expression for the vertical velocity component using a finite difference scheme prescribed in Appendix 10. The unsteady potential flow solution gives us the time dependent value of
Obtain a pseudo tri-diagonal matrix equation to solve the vorticity transport equation in a boundary layer with forward differencing in time and with appropriate differencing in space suitable for
Find the velocity component, in the direction parallel to the surface, by integrating the discrete vorticity values, obtained in Problem 8.2, in the normal direction starting from the wall.Problem
Derive the 2-D vorticity transport equation, and discretize this equation to obtain the vorticity field at time level \(n+1\) using SLUR (Successive Line Underrelaxation).
Obtain the relation between the stream function and the vorticity as the kinematic relation of the 2-D flow. Apply SOR (Successive Overrelaxation) technique to solve the elliptic equation.
What are the differences between the light stall and the deep stall. Comment on the differences as regards the sectional lift and the moment coefficients.
Comment on the effect of the (i) separation,(ii) Mach number on the negative drag for a plunging airfoil.
At high angles of attack, the empirical formulae for the lift and moment coefficients for airfoils pitching at high frequencies are given in terms of maximum dynamic moment coefficient \(\left(C_{M}
During dynamic stall, the drag coefficient is less for pitch-up than for pitch-down, whereas the lift coefficient is larger for pitch-up than for pitch-down. Why?
The indicative of the stall flutter is the sign of the integral under the curve of (i) lift vs vertical displacement for plunging, (ii) moment vs angle of attack for pitching. Why? In obtaining
Using the potential theory obtain the damping for a cycle of (i) plunge, (ii) pitch oscillations.
The state-space representation is based on a state function \(x\) satisfying the first order ODE \(\tau_{1} \dot{x}=x_{o}\left(\alpha-\tau_{2} \dot{\alpha}\right)-x\), where argument
Consider a delta wing with sweep angle \(\Lambda\). Show that the expressions 8.11 and 8.12 give the same lift line slope for the delta wing.Eq 8.11Eq 8.12 = CD CD+CL tan
Using the Polhamus theory obtain the drag polar for a delta wing with sweep angle \(75^{\circ}\).
Obtain the vortex lift line slope, given by Eq. 8.14, for a supersonic delta wing.Eq. 8.14 Ky = -(cos A) ?
A delta wing has an aspect ratio of 1. (i) Plot the coefficients \(K_{p}\) and \(K_{v}\) with respect to Mach number (ii) for \(M=2\), plot the lift coefficient wrt angle of attack. Comment on the
The delta wing given in Fig. 8.59 has the supersonic lift line slope, according to Puckett and Stewart, as follows\[\begin{aligned}& \frac{d C_{l}}{d \alpha}=\left(2 \pi \cot \Lambda /
The induced drag coefficient of the delta wing given in Fig. 8.59, according to Puckett and Stewart, reads as\[C_{D i}=\alpha C_{L}\left[1-m^{\prime} /\left(2(1-a) H(a)
The delta wing given in Example 8.3 is in yaw oscillating with \(35^{\circ}\) amplitude and \(0.40 \mathrm{~s}\) period. Using the coefficients given for yawing moment and \(C_{2}=0.003\)
Derive the formula \(8.20 \mathrm{a}, \mathrm{b}\) and 8.21 which give the effective angle of attack and the effective yaw angle in terms of the yaw angle \(\varphi\).Eq 8.20(a,b)Eq 8.21 CR = a1+a33+
For the wing given in Example 8.3, evaluate (i) the maximum normal force Coefficient, (ii) minimum side force coefficient.Example 8.3At \(25^{\circ}\) angle of attack, a wing with \(80^{\circ}\)
Comment on the aerodynamic mechanisms causing the wing rock of the round leading edged non-slender wings.
Comment on the causes of different types of wing rock and the differences of the period durations involved.
Using the 'Moving wall effect', comment on the negative damping for (i) the plunging profile, (ii) the pitching profile, (iii) the periodically rotating cylinder in a free stream.
Obtain the sectional leading edge suction force coefficient for a profile plunging with \(z_{a}=h \cos k s\) in a free stream at zero angle of attack.
Obtain the expression which gives the wake vortex sheet strength, Eq. 8.29, for Problem 6.25.Eq. 8.29Problem 6.25Using the 'Moving wall effect', comment on the negative damping for (i) the plunging
Show that for an heaving-plunging airfoil the aerodynamic propulsion efficiency is \(\eta=\frac{F^{2}+G^{2}}{F}\).
The unsteady boundary layer solution based on the edge velocity values gives us the skin friction distribution for a body. Obtain the upper and lower boundary layer edge velocity expressions for a
Derive Eq. 8.37 for the thrust coefficient of an airfoil in pure pitching about the point \(a\) with reduced frequency \(k\).Eq. 8.37 de tan (- h+da (1) cos(x(t))) + (1) U da(t) sin(x(t))'
A thin airfoil is plunging with \(h=\bar{h} e^{i w t}\), and pitching with \(\alpha=\bar{\alpha} e^{i(\omega t+\phi)}\) about a point \(a\). Obtain the general expression for the leading edge suction
Derive the thrust efficiency formula, Eq. 8.38, for a pitching plunging airfoil. Comment on the effect of the ratio of the plunge to pitch amplitude on the efficiency.Eq. 8.38 k - k (ka) = x(F + G)
Obtain the time variation of the lift and propulsive force coefficients and their plots for the airfoil given by Example 8.5. Assume that the profile pitches about quarter chord point.Example
Write down a numerical solution algorithm for the LU decomposition solution of the pseudo penta diagonal matrix equation given by (A11.4).A11.4 || B C 2 A2 B2 C2 922 An-1 Bn-1 Cn-1 -1 An Bn R R Rn-1
Obtain Eq. 8.66 as a relation between the wake vorticity and the reduced circulation.Eq. 8.66 w(x) = k(Q; cosx-22, sin x) cos ks +k(Q; sin x+Q, cos x) sinks, Q = Q, +Q;i
Write a numerical algorithm to solve Eq. 8.71 for \(A_{n}\). Obtain lift coefficients for \(h / 2 b=2\) and \(\mathrm{AR}=6,10\) at \(\mathrm{AoA}=12\).Eq. 8.71 Asin no 1 (y) 1 ban b 4h2b2 (cos 0-cos
Using (8.76) find the chord wise variation of the lift as \(L^{\prime}(x)\).Eq 8.76 1 za 2hg ax ax ax - (x)- aza
Using the energy equation, obtain the linearized form of the relation between the perturbation potential and speed of sound, Eq. 6.1.Eq. 6.1 a - a 7-1 t +U) x
Based on the order of magnitude analysis, show that the time derivative of the perturbation potential is negligible compared to \(\mathrm{x}\) derivative.
Obtain Eq. 6.11 from 6.10 by inverse Laplace transform using the convolution integral.Eq. 6.11Eq. 6.10 "(z=0+) = 0 x fo exp(-10) 10 (10)w(x-5)d -1/2 exp(- eo 2eo' 2eo
Show that at constant angle of attack the perturbation potential is given by Eq. 6.13.Eq. 6.13 0(z = 0+) 1 U = 1/2 2e fo - exp(-)[lo() +11(5)], : fo -x = 2e0
Show that at constant angle of attack the surface pressure coefficient is given by Eq. 6.14.Eq. 6.14 Co 84 -2e0 1/ exp(-)10(5)
Show that lower surface pressure is given by Eq. 6.15.Eq. 6.15 Co 8. 848 = 2e0 1/2 exp(-)10(5)
Show that the sectional lift coefficient depends on the angle of attack as expressed in Eq. 6.17.Eq. 6.17 c=4x(fob)-1/2/2 exp(-b) [10(b)+1(b)], 5=2 fob eo
Obtain the sectional moment coefficient and center of pressure using Eq. 6.16.Eq. 6.16 AC -1/2 = 4e0exp(-)o()
Compare the surface pressure coefficient obtained with Dowell method using \(x_{0}=\mathrm{b}\) as expansion point and \(e_{0}=0.12\) and \(f_{0}=2.4 / b\) with the pressure coefficient obtained
Solve Eq. 6.21 in Laplace domain, and obtain the expression 6.22 by inverse transform to give the boundary condition.Eq. 6.21Eq. 6.22 - 22
Obtain Eq. 6.23 from 6.24 by the limiting procedure as \(M_{\infty} \rightarrow 1\).Eq. 6.23Eq. 6.24 (z = 0) = he 2e + 1/2 d eo exp(-ex)x() + -ex)x[L(x) 1 (x)] e
Show that for simple harmonically heaving plunging thin airfoil the surface pressure expression is given by Eq. 6.25 as the free stream Mach number approaches 1.Eq. 6.25 Cp = (nfo2b)/ [2k(x/2b)/ +
Using the values of Example 6.1, plot the phase lag of the surface pressure coefficient along the chord for a heaving plunging thin airfoil.Example 6.1Find the amplitude of the surface pressure for a
Find the amplitude of the (i) sectional lift coefficient, (ii) the sectional moment coefficient about the leading edge using the data given in Example 6.1.Example 6.1Find the amplitude of the
Using Eq. 2.15 expressed for the velocity potential, obtain Eq. 6.26 for compressible steady flows.Eq. 2.15Eq. 6.26 0 = ) A.b+ 1 (8 a ( q + 12 -
What is a 'shock doublet' in unsteady transonic flow?
Discuss the 'transonic dip' phenomenon for the swept wing in a transonic unsteady flow.
What is the function of transonic dip in unsteady transonic flow?
Comment on the 'area rule' for the wing fuselage in transonic flow.
For a given free stream Mach number M, obtain the stagnation pressure coefficient in terms of M. Find the limit of the stagnation pressure coefficient for \(\mathrm{M}\) approaching infinity.
Use the point at the surface and the first point \(\Delta\) n away from the surface, obtain Eqs. 7.11 from 7.10 which gives the pressure caused by the curvature. (take \(\mathrm{R}=-1 /(\mathrm{d}
Using Newton-Busemann theory obtain the surface pressure distribution of a circular cylinder at free stream speed of \(\mathrm{M}=8\), and compare the results with Newton and improved Newton method.
Find the surface pressure distribution of a 2-D body whose surface equation is given with \(z_{u}=x^{1 / 2}, 0 \leq x \leq 1\), and pitching about its leading edge at angle of attack amplitude of
A body of revolution is given with equation of radius \(\mathrm{r}: \mathrm{x}=0.79 \mathrm{r}^{2}-1,-1\) \(\leq x \leq 0\), with respect to symmetry axis \(\mathrm{x}\). Obtain the surface pressure
Find the surface pressure distribution of the body of Problem 7.5 which oscillates about its nose with angle of attack amplitude of \(5^{\circ}\) and \(\mathrm{k}=0.4\).Problem 7.5Obtain the surface
Obtain Eq. 7.6 for steady flows using Eq. 7.20 which expresses surface pressure distribution.Eq. 7.6Eq. 7.20 Cp 2 sin 0,
Newtonian impact theory is non linear. Why?
Find the sectional drag coefficient for a circular cylinder in hypersonic flow using Newton-Busemann theory.
Obtain Eq. 7.24 from 7.21 with expanding the equation into the series in terms of powers of w/a.Eq. 7.21Eq. 7.24 || 22 7-1w 2 a 1+
For a 5% thick airfoil find the free stream Mach number for which: (i) first order, (ii) second order (iii) third order piston theory is applicable.
What should be the minimum value of reduced frequency \(\mathrm{k}\) for a thin airfoil in a free stream Mach number of 6 so that the piston theory is valid?
Why is it necessary to consider the thickness effect for a thin airfoil in hypersonic aerodynamics?
Find the surface pressure distribution of a 5% thick parabolic airfoil in a free stream Mach number of 5 at altitude of \(50 \mathrm{~km}\).
Find the lifting pressure distribution for the profile given in Problem 7.16 at free stream Mach number of \(\mathrm{M}=3\).Problem 7.16The profile given in Problem 7.12. is in plunging motion with
Obtain the oblique shock relations in terms of Mach number M, shock angle \(\theta_{\mathrm{s}}\), the specific heat ratios \(\gamma\).
Show that the enthalpy can be expressed in terms of the pressure and the density as follows: \(\mathrm{h}=2 \gamma /(\gamma-1)(\mathrm{p} / ho)\).
Find the pressure, density, enthalpy and the surface velocity of Example 7.2 for the streamline value of \(\chi=0.0625\).Example 7.2For a spherical shock with radius of 1 immersed in \(M=\infty\) and
Use the Maslen method to find the flow conditions at a point with seen with \(30^{\circ}\) angle from the center of a circular cylinder in free stream Mach number of infinity.
Show that the Maslen method gives: \(\tan \theta_{\mathrm{s}}=\left[11-\left(121-48 \tan ^{2} \theta_{\mathrm{c})}^{1 / 2}\right] / 2 \tan \theta_{\mathrm{c}}\right.\) for the the shock angle
What is the advantage of the Maslen method over the Newtonian impact theory?
Using the Maslen method find the change in the shock angle at the nose of a cone pitching with about its nose with a small amplitude. Use body attached coordinate system as done in Example
Obtain the Jacobian fluxes \(A_{i}=\partial F_{i} / \partial U\) in Eq. 7.47 for \(\mathrm{i}=1,2\).
The coefficient \(\lambda=1 / 2 \mathrm{~h}^{2}\left|\partial u_{l} / \partial x_{l}\right|\) is used to prevent the numerical oscillations at a shock. Obtain the expression for \(\left|\partial
An empirical way to determine shock shapes based on experiments is given by Billig. A shape of shock created by a blunt body in x−yx−y coordinate system is given as a hyperbola Extra \left or
A flat plate of \(4 \mathrm{~m}\) long has the wall temperature of \(1200 \mathrm{~K}\) at zero angle of attack at free stream Mach number of 25 and \(85 \mathrm{~km}\) altitude. Using the data given
Find the induced pressure on the surface of cone given in Example 7.3 with the free stream conditions described at Problem 7.31.Example 7.3Analyzing the conic shock about a slender cone at
Obtain aerodynamic heating formula for at the stagnation point of a sphere in terms of radius of curvature. Assuming calorically perfect air, find the aerodynamic heating of the space capsule given
Show that Eq. 7.68 gives the rate of heating in terms of the surface pressure change for the shock boundary layer interaction.Eq. 7.68 Qwx (Pe (Pe) 4/5
For a diatomic molecule find the contribution of the vibration energy to the specific heat under constant volume. Neglect the ground level energy effect to the partition function.
Find the specific internal energy and the enthalpy of the air under 1 atmosphere pressure and \(2000 \mathrm{~K}\).
The surface area of the space shuttle is designed to be 560000in2.560000in2. Determine the lift coefficient of the shuttle during its re-entry.
For \(\mathrm{N}_{2}+\mathrm{O}_{2} \rightarrow 2 \mathrm{~N}+\mathrm{O}_{2}\) plot the graph of forward reaction in \(6000 \mathrm{~K}-9000 \mathrm{~K}\) interval and compare with the production of
In pure N2N2 flow, find the temperature and density change with respect xx coordinate behind the normal shock created with M=12.28, T=300 KM=12.28, T=300 K and
Using the energy equation obtain Eq. 7.95 for the stagnation enthalpy hoho in terms of local Prandtl and Lewis numbers.Eq. 7.95 aci +(1-1/Pr)uu- +(1-1/Le)pD12 hi- z z z Pr z ho pu- = + pw
Find for a waverider flying at \(80 \mathrm{~km}\) altitude with a Mach number of 25, find:(i) approximate lifting pressure, (ii) lift coefficient, and induced drag coefficient. Assume ideal flow.
Find the sectional lift coefficient for a thin symmetric airfoil with integrating the lifting pressure coefficient.
Find the sectional moment coefficient of a thin symmetric airfoil with respect to the mid chord. Then find (i) the center of pressure(ii) the aerodynamic center of the airfoil considered.
Using the approximate expression of the Theodorsen function for the vertical motion of an airfoil given by \(z_{a}(t)=\mathrm{h} \cos (\mathrm{ks})\) where \(s=U t / b\), find the sectional lift
The exact expression for the Theodorsen is \(C(k)=\mathrm{H}_{1}{ }^{2}(k) /\left[\mathrm{H}_{1}{ }^{2}(k)+\mathrm{iH}_{\mathrm{o}}{ }^{2}(k)\right]\). Plot the real and imaginary parts of the
The graph of the lift versus drag coefficient is called the drag polar. Plot a drag polar for a thin wing for incompressible flow.

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