MAE 130C - Midterm 2 Theoretical Questions

Define a Prandtl-Meyer expansion fan and describe how the Mach number and pressure change as flow passes through it.

A continuous, centered fan of infinitesimally weak Mach waves that occurs when a supersonic flow turns around a sharp or smooth convex corner (or expands gradually). The process of the flow passing is isentropic, Mach number increases, and pressure decreases

How does increasing the Aspect Ratio (AR) of a wing affect the induced drag coefficient?

Increasing the aspect ratio of the wing will decrease the induced drag coefficient. This is because a longer wing will spread the same lift over a greater distance, reducing the strength of the tip vortices and the downwash angle. Weaker vortices produce less drag.

• think of the induced drag equation! higher AR, lower CDi, increased efficiency

Using integral methods, if you assume a linear velocity profile u/U = y/δ, calculate the ratio δ∗/δ.

After lots of math, you get 1/2 (must know the equation for displacement thickness and apply known u/U and integrate over the thickness of the B.L)

1. delta*= integral 0→delta (1-u/U)dy, turns into delta integral 0→ 1(1-y/delta)dy

2. we can plane N=y/delta and do substitution before deriving!

3. delta integral 0→1 (1-N) dN = delta *

4. integrating, we get N-N²/2 | 0→1 and plugging in gives us delta * = delta/2.

5. delta*/delta = 1/2

Describe the difference between a Mach wave and an oblique shock wave.

An oblique shock wave is a finite-strength compression wave that turns the flow discontinuously, increasing entropy. A Mach wave is a weak, isentropic disturbance inclined at the Mach angle μ.

Mach Wave	Oblique Shock
infinitely weak disturbance	finite-strength compression
isentropic	non-isentropic
no entropy increase	entropy increases
no total pressure loss	total pressure loss
tiny pressure change	sudden pressure jump

Briefly explain how the circulation Γ is related to the pressure difference between the upper and lower surfaces of an airfoil.

Circulation Γ is a measure of the net "rotation" of flow around the airfoil, which creates a velocity difference across the airfoil, leading via Bernoulli to a pressure difference that integrates to lift.

Explain the concept of linearized supersonic flow theory and state the primary assumption regarding the flow deflection angle.

An assumption that the flow deflection angle is infinitesimally small, which allows the nonlinear gas dynamics equations to be linearized, leading to simple analytical expressions for pressure coefficient and lift/drag on thin airfoils.

Distinguish between skin friction drag and pressure drag. Which component is more significant for a bluff body (e.g., a cylinder) at high Reynolds numbers, and why?

Skin friction drag arises from shear stresses due to viscosity in the boundary layer, while pressure drag results from the pressure distribution caused by flow separation. For a bluff body like a cylinder at high Reynolds numbers, pressure drag is far more significant because extensive flow separation creates a large low-pressure wake, vastly outweighing the relatively small skin friction contribution.

For a supersonic flow over a concave corner, what type of wave is formed, and how do the pressure and temperature change across it?

An isentropic compression wave, pressure and temperature both increase

• PRETTY SURE THE TOP IS WRONG.

• should be oblique shock as it specifies a corner and doesnt specify it being smooth

• pressure and temperature however do still increase

Define the "Momentum Thickness" (θ) and explain how it represents the loss of momentum flux in the boundary layer.

Momentum thickness is the height of freestream flow that has momentum equivalent to the total momentum loss due to the boundary layer. It represents how much the wall "slows down" the flow in terms of momentum flux.

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momentum thickness is a boundary-layer thickness measure that represents the loss of momentum in the boundary layer due to viscous effects. It measures the equivalent thickness of fluid that would have to be removed from an ideal uniform flow to account for the momentum deficit caused by the slower velocities in the boundary layer.

In the boundary layer, fluid near the wall moves slower than the free stream velocity U, so the flow carries less momentum than a uniform flow at U. The momentum thickness integrates this velocity deficit across the boundary layer to quantify the total reduction in momentum flux.

Why is the integral method often preferred over solving the full Navier-Stokes equations for engineering estimates of boundary layer growth?

It reduces the boundary layer problem to an ordinary differential equation (ODE) by integrating the momentum equation across the boundary layer, Also, only integral quantities like momentum thickness (θ) and displacement thickness (δ∗) matter, not the exact velocity distribution u(y)

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The integral method is preferred because it reduces the boundary-layer equations to a simpler ordinary differential equation (ODE) by integrating the momentum equation across the boundary layer. It only requires integral quantities such as momentum thickness (θ)(\theta)(θ) and displacement thickness (δ∗)(\delta^*)(δ∗), rather than solving for the exact velocity profile u(y)u(y)u(y) everywhere in the flow.

How does the stagnation pressure change across an oblique shock compared to a normal shock of the same upstream Mach number?

The stagnation pressure between upstream and downstream of a shock depends on the component of the velocity/Mach number normal to the direction of the shock. Because the normal Mach number of flow crossing an oblique shock is less than that crossing a normal shock, stagnation pressure across an oblique shock changes less

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an oblique shock causes a smaller decrease in stagnation pressure than a normal shock.

This is because only the normal component of the Mach number:

Mn1=M1sin⁡β

is compressed through the oblique shock, making the shock weaker than a normal shock, where the entire Mach number is normal to the shock. Therefore, oblique shocks generate less entropy and have smaller total pressure losses.

Explain the "Momentum Deficit" concept as it relates to the generation of drag on an immersed body

When a viscous fluid flows over an immersed body (like an airfoil, sphere, or flat plate), the no-slip condition at the wall slows down fluid particles near the surface forming a boundary layer downstream of the body. This creates a region of lower streamwise momentum compared to the freestream. The total drag on the body equals the rate of loss of momentum of the fluid.

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Momentum deficit is the loss of flow momentum behind an object because viscosity slows the fluid near the surface of the body. The wake behind the object contains slower-moving fluid than the free-stream flow, so the outgoing flow carries less momentum than the incoming flow. This loss of momentum creates a drag force on the body.

Describe the reflection of an oblique shock wave from a solid, frictionless wall.

The wave is reflected as another oblique shock. Since the wall is frictionless (inviscid), only the flow direction boundary condition matters: the flow must be tangent to the wall both before and after the interaction.

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The oblique shock reflects off the solid, frictionless wall as a reflected wave. In the case of regular reflection, this wave is another oblique shock that turns the flow so that the downstream velocity is again parallel to the wall. Because the wall is inviscid, the only boundary condition is zero normal velocity at the wall.

State the Kutta-Joukowski theorem and explain the physical relationship between circulation Γ and the lift per unit span L'.

The lift per unit span is directly proportional to circulation, with ρ∞V∞ as the proportionality factor. This is because The total downward momentum flux imparted to the air per unit time equals the lift. That momentum flux is directly related to Γ, and more circulation means more of a pressure difference and more lift.

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The Kutta–Joukowski theorem states that the lift per unit span on a two-dimensional airfoil in a steady, incompressible, inviscid flow is given by:

L′=ρV∞Γ

where Γ is the circulation around the airfoil.

Lift is directly proportional to circulation. Circulation represents the net “rotation” of the flow around the airfoil, which creates a velocity difference between the upper and lower surfaces. This velocity difference leads to a pressure difference, producing lift. More circulation → stronger pressure difference → more lift.

What is the effect of downwash velocity on finite width wing?

Downwash from wingtip vortices tilts the local flow downward, reducing the effective angle of attack and generating induced drag. The finite wing produces less lift for a given geometric α and requires more thrust to overcome induced drag compared to an infinite wing at the same CL​

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Downwash from wingtip vortices tilts the local airflow downward, reducing the effective angle of attack and tilting lift backward, which generates induced drag and reduces aerodynamic efficiency.

Describe the "Shape Factor" (H = δ ∗/θ) and explain its utility in characterizing the state of a boundary layer.

The shape factor H=δ∗/θ quantifies the "fullness" of a boundary layer velocity profile, with high values indicating a profile prone to separation and low values indicating a fuller, more robust profile. Its utility lies in providing a simple criterion to assess boundary layer health: as H rises above ~3.5 for laminar flow or ~2.5 for turbulent flow, separation is imminent, making it a key parameter in integral methods for drag and separation prediction.

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the shape factor H=δ∗/θ measures the fullness of a boundary layer velocity profile. Higher values indicate a “slower near-wall” profile (laminar or near separation), while lower values indicate a fuller, well-mixed turbulent boundary layer with higher near-wall momentum.

Explain the Kutta-Joukowski Theorem and how it allows for the calculation of lift per unit span.

The lift per unit span L′ on a two-dimensional body in steady inviscid flow is directly proportional to the circulation Γ around it: L′=ρ∞V∞Γ. This allows lift calculation by first determining Γ from the flow conditions and the Kutta condition (which fixes the circulation by requiring smooth flow off the trailing edge).

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The Kutta–Joukowski theorem states that lift per unit span is proportional to circulation: L′=ρV∞Γ It allows lift to be calculated by finding the circulation around the airfoil, which represents the net rotation of the flow that creates a pressure difference between the upper and lower surfaces.

At what point in a boundary layer does flow separation occur, defined in terms of the wall shear stress?

Flow separation occurs at the point where the wall shear stress becomes zero

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Flow separation occurs when the wall shear stress drops to zero (and changes sign), meaning the velocity gradient at the wall vanishes and the near-wall flow reverses direction.

How is a weak (acoustic) disturbance felt in supersonic flow?

In supersonic flow, a weak (acoustic) disturbance propagates only within a conical region known as the Mach cone, bounded by the Mach angle μ=arcsin⁡(1/M). Outside this cone, the flow is completely unaware of the disturbance, meaning information cannot travel upstream.

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n supersonic flow, a weak disturbance is convected downstream along Mach waves at the Mach angle μ=sin⁡−1(1/M), forming a Mach cone, and cannot influence the upstream flow.

Define the Mach angle and explain its physical significance in a supersonic flow.

The Mach angle μ=arcsin⁡(1/M) is the half-angle of the conical surface across which weak pressure disturbances propagate in a supersonic flow. Physically, it defines the boundary between the region that can "feel" a disturbance and the region that cannot, since no information can travel upstream in supersonic flow.

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The Mach angle is the angle of propagation that defines the region of influence of weak disturbances in supersonic flow. It is a half angle of the Mach cone and defines the boundary between the region influenced by a disturbance and the region that is unaffected, since information cannot travel upstream in supersonic flow.

What is the relationship between the momentum thickness θ and the drag force on a flat plate of length L?

The momentum thickness θ at the trailing edge of a flat plate is directly related to the drag force per unit width by the equation D=ρU∞^2θ(L), where ρ is density and U∞ is the free-stream velocity. Thus, measuring the deficit in momentum flux within the boundary layer at x=L provides the total skin friction drag on the plate.

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The drag per unit span is proportional to the momentum thickness at the trailing edge:

D′=ρU∞2θ(L)D' = \rho U_\infty^2 \theta(L)D′=ρU∞2​θ(L)

This means that momentum thickness measures the momentum lost due to the boundary layer, so a larger θ\thetaθ at the end of the plate means greater skin-friction drag.

Briefly describe the starting vortex phenomenon. How does Kelvin's theorem explain the development of circulation around an airfoil that starts from rest?

When an airfoil starts from rest, the viscous boundary layer causes a starting vortex to be shed from the trailing edge into the wake, which induces an equal and opposite circulation around the airfoil. Kelvin's theorem states that the circulation around a closed fluid loop remains constant, so the initial zero circulation is preserved by the formation of the starting vortex with circulation equal in magnitude but opposite in sign to that developing around the airfoil.

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The starting vortex is a vortex shed into the wake when an airfoil starts from rest. By Kelvin’s theorem, total circulation must remain zero, so the development of bound circulation around the airfoil is balanced by an equal and opposite starting vortex in the wake

Explain why wall shear stress τw is related to the velocity gradient at the wall for a Newtonian fluid.

For a Newtonian fluid, the shear stress is proportional to the rate of strain, and at the wall, the no-slip condition forces the fluid velocity to change rapidly from zero to the free-stream value over a short distance. This results in the wall shear stress being directly proportional to the velocity gradient perpendicular to the wall

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Because in a Newtonian fluid, viscous shear stress is proportional to the rate of deformation (velocity gradient). The no-slip condition makes the fluid velocity change from zero at the wall to a finite value above it, creating a gradient. Viscosity resists this gradient, producing shear stress:

Contrast the propagation of acoustic disturbances in subsonic versus supersonic flows

In subsonic flows, acoustic disturbances propagate in all directions and can travel upstream, allowing the flow field to adjust to downstream conditions. In supersonic flows, disturbances are confined within a Mach cone downstream of the source, as the flow velocity exceeds the speed of sound, preventing any upstream influence.

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• Subsonic (M < 1): Acoustic disturbances travel in all directions, including upstream, so the entire flow field can be influenced.

• Supersonic (M > 1): Disturbances are confined to a downstream Mach cone and cannot travel upstream, so upstream flow is unaffected.

Define "Circulation" (Γ) and explain its mathematical relationship to the velocity field around a closed curve.

Circulation (Γ) is defined as the line integral of the velocity vector around a closed curve. It quantifies the net rotational tendency of the fluid within the curve and is related to vorticity via Stokes' theorem

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Γ=∮C​V⋅dl

Circulation is the line integral of velocity around a closed curve. It measures the net rotational (swirling) effect of the flow around the loop, based on how much the velocity is aligned with the path direction as you move around it.

Describe the impact of an "Adverse Pressure Gradient" (dp/dx>0) on flow separation.

An adverse pressure gradient (dp/dx > 0) causes the flow near the wall to decelerate, as pressure increases in the streamwise direction and reduces the fluid's kinetic energy. This deceleration can lead to flow separation when the near-wall momentum is insufficient to overcome the pressure rise, causing the boundary layer to detach from the surface and form a reverse-flow region.

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An adverse pressure gradient (dp/dx>0dp/dx > 0dp/dx>0) means pressure increases in the direction of flow, which opposes the motion. This slows down the fluid in the boundary layer, especially near the wall, reducing its momentum. If the fluid loses too much momentum, it can no longer move forward against the rising pressure, causing the flow to reverse and separate from the surface.

What is the maximum deflection angle θmax for a given Mach number, and what happens to the shock wave if the wedge angle exceeds this value?

The maximum deflection angle θmax for a given Mach number is the largest angle that an attached oblique shock can turn the flow; if the wedge angle exceeds θmax, the shock wave detaches from the wedge tip and becomes a curved bow shock positioned upstream.

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or a given upstream Mach number, there is a maximum flow deflection angle θmax⁡\theta_{\max}θmax​ for which an attached oblique shock can exist on a wedge.

If the wedge angle is less than or equal to θmax⁡\theta_{\max}θmax​:

• An oblique shock attaches to the wedge tip.

If the wedge angle exceeds θmax⁡\theta_{\max}θmax​:

• An attached oblique shock is no longer possible.

• The shock detaches and forms a curved bow shock in front of the wedge, and the flow is turned through a stronger, detached shock structure instead.

Define the lift coefficient CL and drag coefficient CD using non-dimensional parameters.

The lift coefficient CL and drag coefficient CD​ are dimensionless parameters used to quantify the lift and drag forces acting on a body. They are defined as CL=L/(1/2ρU∞^2)A)​ and CD=D/(1/2ρU∞^2A)​, where L is lift force, D is drag force, ρ is fluid density, U∞​ is free-stream velocity, and A is a reference area (typically planform area for wings or frontal area for bluff bodies)

What is a "slipstream," and under what conditions does it form during shock wave interactions?

A thin, high shear boundary layer between two streams at different velocities. This occurs under steady, inviscid, supersonic flow conditions where the pressure is equal across the interface but the velocities, densities, and temperatures differ. It forms when three shocks interact in a Mach reflection, separating flows that passed through different shock paths.

Define the "Displacement Thickness" (δ∗) and provide its physical interpretation in terms of mass flow rate.

The displacement thickness (δ*) is defined as the distance the external flow is displaced outward from the wall due to the velocity deficit in the boundary layer, and physically it represents the reduction in mass flow rate caused by the boundary layer, equivalent to the thickness of an inviscid flow region carrying the same mass flow deficit.

Discuss how Kelvin's Circulation Theorem ensures that the total circulation of the system (airfoil plus starting vortex) remains zero

As an airfoil starts from rest in an initially irrotational flow (zero total circulation), the shedding of a starting vortex with clockwise circulation forces the airfoil to develop an equal and opposite counterclockwise circulation, ensuring the total circulation of the system (airfoil plus starting vortex) remains zero at all times.

Explain the difference between skin friction drag and pressure drag (form drag). Which is dominant for a streamlined body?

Skin friction drag arises from viscous shear stresses acting tangentially along the surface, while pressure drag results from the normal pressure distribution due to flow separation and wake formation. For a streamlined body, skin friction drag is typically dominant because the gradual shape maintains attached flow, minimizing pressure drag.

In the Von Karman equation, what does the term involving the pressure gradient (dp/dx) represent physically?

In the Von Karman momentum integral equation, the term involving the pressure gradient (dp/dx) physically represents the effect of the streamwise pressure variation on the rate of change of momentum thickness, accounting for how an adverse or favorable pressure gradient modifies boundary layer growth and wall shear stress.

How does the total drag of a sphere change as the boundary layer transitions from laminar to turbulent?

As the boundary layer on a sphere transitions from laminar to turbulent, the total drag coefficient drops sharply (the "drag crisis") due to delayed flow separation, resulting in a narrower wake and reduced pressure drag despite increased skin friction.

Describe D'Alembert's Paradox and explain why it does not hold true in real-world fluid flows.

For an inviscid, incompressible, irrotational flow, the drag force on a body moving at constant velocity is zero, which contradicts physical observation. In real-world flows, viscosity is always present, causing skin friction drag and, more importantly, boundary layer separation that leads to pressure drag due to asymmetric pressure distribution and wake formation, so drag is never zero.

Using the β − θ − M relation, explain what happens to the shock angle β if the deflection angle θ is increased while the upstream Mach number remains constant.

As the deflection angle θ increases from zero, the shock angle β initially increases from the Mach angle μ (where μ = arcsin(1/M)) up to a maximum β_max at the maximum possible θ (θ_max)

Explain the difference between "strong" and "weak" oblique shock solutions. Which one is typically observed in unconfined external flows?

The strong oblique shock solution yields a higher shock angle and subsonic downstream flow, while the weak solution yields a lower shock angle and generally supersonic downstream flow; in unconfined external flows, the weak shock solution is typically observed because it satisfies the downstream boundary conditions with lower entropy rise.

Describe the "Downwash" effect caused by wing-tip vortices on a finite wing.

Downwash is the downward deflection of the airflow behind a finite wing caused by wing-tip vortices, which reduces the effective angle of attack and generates induced drag.

Is the flow through a Prandtl-Meyer expansion fan isentropic? Justify your answer.

The flow through a Prandtl-Meyer expansion fan is isentropic because the expansion occurs through a continuous series of infinitesimally weak Mach waves, resulting in reversible, adiabatic processes with no shock-induced entropy rise.

For a laminar flat-plate boundary layer (Blasius solution), how does the boundary layer thickness δ scale with the Reynolds number Rex?

For a laminar flat-plate boundary layer in the Blasius solution, the boundary layer thickness δ scales as δ~(5.0x)/sqrt(Rex)

What is the "Skin Friction Coefficient" Cf , and how is it defined in terms of τw?

The skin friction coefficient Cf is a dimensionless parameter that quantifies the local wall shear stress τw relative to the dynamic pressure of the free stream, defined as Cf=τw/(1/2ρU∞^2), where ρ is fluid density and U∞ is free-stream velocity.

Why does a turbulent boundary layer generally resist flow separation better than a laminar boundary layer?

A turbulent boundary layer generally resists flow separation better than a laminar boundary layer because its chaotic mixing and higher momentum exchange near the wall replenish low-momentum fluid with higher-momentum fluid from the outer region, enabling it to overcome adverse pressure gradients more effectively.

Define the pressure coefficient Cp in the context of linearized supersonic theory.

The pressure coefficient is a non-dimensional representation of the pressure change from a reference state. It is defined as Cp=(2θ)/sqrt(M^∞2−1) for small flow deflections θ, where M∞​ is the free-stream Mach number

How does the wave drag of a thin airfoil in supersonic flow scale with the thickness-to-chord ratio according to linearized theory?

The wave drag coefficient of a thin airfoil scales with the square of the thickness-to-chord ratio (t/c)², while also being proportional to 1/sqrt(M∞^2−1)

What are the two primary physical sources of aerodynamic drag on a body in a viscous, incompressible flow?

The two primary physical sources of aerodynamic drag in a viscous, incompressible flow are skin friction drag, caused by shear stresses acting tangentially on the surface, and pressure drag (form drag), caused by the net pressure imbalance from flow separation and wake formation.

State the Kutta Condition and explain its role in determining the circulation around an airfoil with a sharp trailing edge.

The Kutta condition states that for an airfoil with a sharp trailing edge, the flow leaves the trailing edge smoothly with equal pressure on both upper and lower surfaces, forcing the rear stagnation point to coincide exactly with the trailing edge. This condition uniquely determines the circulation around the airfoil by ensuring that the velocity remains finite at the trailing edge, which otherwise would be infinite in potential flow theory.

Write the Von Karman Integral Momentum Equation for a 2D incompressible boundary layer.

(dθ/dx)+(2+H)(θ/Ue)(dUe/dx)=Cf/2

Define "Stall" in the context of an airfoil and describe the behavior of the boundary layer during this event.

Stall is a sudden loss of lift on an airfoil caused by exceeding the critical angle of attack, where the boundary layer separates massively from the upper surface due to an adverse pressure gradient. During stall, the separated boundary layer forms a large recirculating wake, drastically reducing lift and increasing pressure drag.

What is "Induced Drag" (or lift-induced drag), and how does it differ from profile drag?

Induced drag is the drag generated by the trailing vortices created by lift on a finite wing, whereas profile drag is the sum of skin friction and pressure drag from the airfoil's cross-section in 2D flow (or from an infinite wing).

Compare the velocity profiles of a laminar boundary layer and a turbulent boundary layer. Which profile has a higher velocity gradient at the wall?

A laminar boundary layer has a smooth, parabolic-like velocity profile with a lower gradient at the wall, while a turbulent boundary layer has a fuller, more energetic profile with a much steeper velocity gradient near the wall due to intense mixing. Consequently, the turbulent boundary layer has a higher velocity gradient at the wall.

Describe the impact of a "Favorable Pressure Gradient" (dp/dx < 0) on the stability of a boundary layer.

A favorable pressure gradient (dp/dx < 0) accelerates the flow, stabilizing the boundary layer by reducing inflectional instabilities and delaying transition from laminar to turbulent flow. This promotes a thinner, more attached boundary layer with lower tendency for separation.

Explain the concept of a "Bluff Body" and why its drag is dominated by pressure drag.

A bluff body is a non-streamlined shape (e.g., a sphere, cylinder, or flat plate normal to flow) that causes extensive flow separation, creating a large low-pressure wake behind it. This wake produces a significant pressure imbalance between the front and rear surfaces, making pressure drag dominate over skin friction drag.

For a supersonic diamond-shaped airfoil, describe the wave pattern (shocks and expansions) at the leading and trailing edges.

Two oblique shock waves form—one from the upper apex and one from the lower apex—compressing and turning the flow; at the trailing edge, two expansion fans (Prandtl-Meyer expansions) occur from the upper and lower surfaces downstream of the shock waves, turning the flow back to the free-stream direction

In supersonic flow, does a thicker airfoil generally produce more or less lift-induced drag compared to a thinner airfoil of the same planform?

In supersonic flow, a thicker airfoil generally produces more lift-induced (wave) drag due to stronger shock waves and higher wave drag, whereas lift-induced drag from vorticity is negligible compared to wave drag.

How does the Mach angle change as the Mach number approaches 1.0 (transonic limit)?

As the Mach number approaches 1.0 (the transonic limit), the Mach angle μ, defined as μ=arcsin⁡(1/M), increases toward 90°, meaning weak disturbances propagate almost perpendicular to the flow direction.

Define the "Region of Influence" and "Zone of Silence" for a point source moving at supersonic speed.

For a point source moving at supersonic speed, the "region of influence" is the downstream cone (Mach cone) within which disturbances from the source can be felt, while the "zone of silence" is the region outside this cone where no disturbances reach because the source travels faster than the waves it emits.