ASEN 3711 Exam 1 High Yield

[Exam | Forces] What two surface distributions create all aerodynamic forces and moments on a body?

Pressure distribution (normal to the surface) and shear-stress distribution (tangent to the surface).

[Exam | Forces] In what direction does pressure act on a surface?

Normal (perpendicular) to the surface.

[Exam | Forces] In what direction does viscous shear stress act on a surface?

Tangential (parallel) to the surface.

[Exam | Forces] How are lift and drag oriented relative to the freestream?

Lift is perpendicular to the freestream; drag is parallel to the freestream.

[Exam | Terms] What is the divergence of velocity written in vector notation?

∇·V.

[Exam | Terms] What is the temperature gradient written in vector notation?

∇T.

[Exam | Terms] What is the physical meaning of ∇×V?

The vorticity vector at that location in the flow.

[Exam | Terms] What does ∭_V ρg dV represent?

The weight (body force due to gravity) of the fluid inside the control volume.

[Exam | Terms] What does ∯_S (ρV·dS)V represent?

The net flux of momentum out of the control volume.

[Exam | Terms] What does -∮_C V·ds represent under the course sign convention?

Circulation Γ about the closed curve C.

[L02 | Gradient] What does the gradient of a scalar field represent?

A vector pointing in the direction of greatest increase; its magnitude is the maximum directional rate of increase.

[L02 | Gradient] How is ∇p oriented relative to constant-pressure contour lines or surfaces?

It is normal (perpendicular) to the constant-p contours/surfaces.

[L02 | Divergence] What does divergence physically measure?

Net outward flux per unit volume, or the local tendency of a vector field to expand from a point.

[L02 | Divergence] Write ∇·V in Cartesian coordinates.

∂u/∂x + ∂v/∂y + ∂w/∂z.

[L02 | Curl] What does curl physically measure?

The local rotational tendency of a vector field; for velocity it is vorticity.

[L02 | Surface integral] What does ∯_S A·dS measure?

Net flux of vector A through the surface.

[L02 | Stokes] State Stokes’ theorem.

∮_C A·ds = ∬_S (∇×A)·dS; circulation around a boundary equals curl flux through the enclosed surface.

[L02 | Divergence theorem] State the divergence theorem.

∯_S A·dS = ∭_V ∇·A dV; outward surface flux equals the volume integral of divergence.

[L03 | Frames] What is an Eulerian frame of reference?

A fixed spatial/control-volume viewpoint; fluid can flow through the region.

[L03 | Frames] What is a Lagrangian or material frame of reference?

A viewpoint that follows the same material fluid particles as they move.

[Exam | Frames] True or false: An Eulerian frame is a material frame that follows the fluid.

False. Eulerian is fixed in space; Lagrangian follows the material.

[L03 | Integral vs differential] What do integral conservation laws describe?

Bulk or average behavior over a finite control volume.

[L03 | Integral vs differential] What do differential conservation laws describe?

Local variations at an infinitesimal fluid element.

[L03 | Mass] State conservation of mass in words.

Mass can be neither created nor destroyed.

[L03 | Mass] State the integral continuity equation for a fixed control volume.

∂/∂t ∭_V ρ dV + ∯_S ρV·dS = 0.

[L03 | Mass terms] What does ∂/∂t ∭_V ρ dV represent?

The time rate of accumulation of mass inside the control volume.

[L03 | Mass terms] What does ∯_S ρV·dS represent?

Net mass flow rate out through the control surface.

[L03 | Sign] Why is inflow negative in a closed-surface flux integral?

The outward surface normal points opposite the inward velocity, so V·dS < 0.

[L03 | No-flux surface] When is mass flux through a control surface zero?

When the velocity is tangent to the surface, so V·n = 0.

[L03 | Mass] State the differential continuity equation.

∂ρ/∂t + ∇·(ρV) = 0.

[L03 | Incompressible] What does incompressible flow imply for continuity?

Density of each fluid element is constant and ∇·V = 0.

[L03 | 1D incompressible] What is the steady incompressible duct relation?

A₁V₁ = A₂V₂.

[L03 | Momentum] State the integral momentum equation used in lecture.

∂/∂t ∭_V ρV dV + ∯_S (ρV·dS)V = -∯_S p dS + ∭_V ρf dV + F_visc.

[L03 | Momentum terms] What does ∯_S (ρV·dS)V represent?

Net convective momentum flux out of the control volume.

[L03 | Momentum forces] What are the two main classes of force on a control volume?

Body forces acting throughout the volume and surface forces acting on the boundary.

[L04 | Visualization] Define a streakline.

The line connecting all particles that have passed through a specified point.

[L04 | Visualization] Define a pathline.

The actual trajectory of one fluid particle through time.

[L04 | Visualization] Define a streamline.

A curve that is tangent to the instantaneous velocity vector everywhere.

[Exam | Visualization] In a steady flow, which of streaklines, pathlines, and streamlines coincide?

All three coincide in a steady flow.

[L04 | Streamline equation] What differential equation defines a 2D streamline?

dy/dx = v/u.

[L04 | Streamline arrows] How do you determine arrow direction on a plotted streamline?

Evaluate the local velocity components and point the arrow in the direction of V.

[L04 | Substantial derivative] State the substantial/material derivative of a scalar property f.

Df/Dt = ∂f/∂t + V·∇f.

[L04 | Substantial derivative] What does the local term ∂f/∂t measure?

Change with time at a fixed spatial location (unsteadiness).

[L04 | Substantial derivative] What does the convective term V·∇f measure?

Change experienced because a moving particle travels through spatial gradients.

[L04 | Substantial derivative] What does Df/Dt measure physically?

The total time rate of change experienced by a material fluid element moving through space and time.

[Exam | Substantial derivative] In D(ρV)/Dt = ∂(ρV)/∂t + V·∇(ρV), which term is the local rate of change?

∂(ρV)/∂t.

[Exam | Substantial derivative] In D(ρV)/Dt = ∂(ρV)/∂t + V·∇(ρV), which term is the convective rate of change?

V·∇(ρV).

[Exam | Substantial derivative] In D(ρV)/Dt = …, which expression is the rate experienced by a material element?

The full substantial derivative D(ρV)/Dt.

[L04 | Steady vs acceleration] Can a steady flow have nonzero material acceleration?

Yes. The local term is zero, but convective acceleration can be nonzero.

[L04 | Dilatation] What is dilatation?

The fractional rate of change of a material fluid element’s volume.

[L04 | Vorticity] Define vorticity.

ξ = ∇×V.

[L04 | Angular velocity] How is fluid-element angular velocity related to vorticity?

ω = ½ξ = ½(∇×V).

[L04 | Irrotational] What does irrotational flow mean?

Vorticity is zero: ∇×V = 0.

[L04 | Circulation] State the course definition of circulation.

Γ = -∮_C V·ds.

[L04 | Circulation sign] Under the lecture sign convention, what direction corresponds to positive Γ?

Clockwise circulation.

[L04/L05 | Stream function] Under what condition does a 2D stream function ψ exist?

For a two-dimensional incompressible flow.

[L04/L05 | Stream function] Relate ψ to Cartesian velocity components.

u = ∂ψ/∂y and v = -∂ψ/∂x.

[Exam | Components] What is ∂φ/∂x?

u.

[Exam | Components] What is ∂φ/∂y?

v.

[Exam | Components] What is ∂ψ/∂y?

u.

[Exam | Components] What is -∂ψ/∂x?

v.

[L04/L05 | Polar stream function] Relate ψ to polar velocity components.

Vᵣ = (1/r)∂ψ/∂θ and Vθ = -∂ψ/∂r.

[L04/L05 | Streamline] What is constant along a streamline in 2D incompressible flow?

The stream function ψ.

[L04/L05 | Stream function meaning] What does the difference Δψ between two streamlines represent?

Volumetric flow rate per unit span between the streamlines.

[L04/L05 | Streamline crossing] Can fluid cross a streamline?

No; velocity is tangent to it, so normal velocity is zero.

[L04/L05 | Velocity potential] Under what condition does a velocity potential φ exist?

For irrotational flow.

[L04/L05 | Velocity potential] Relate φ to velocity.

V = ∇φ.

[L05 | Path independence] Why is the velocity-potential integral path independent?

Irrotational flow has zero circulation around any closed loop, so the line integral between two points is independent of path.

[L05 | Orthogonality] How are streamlines and equipotential lines oriented in 2D potential flow?

They intersect at right angles: ψ-lines are perpendicular to φ-lines.

[L05 | Solving ψ/φ] When integrating one velocity relation to find ψ or φ, why must you add an unknown function of the other variable?

Partial integration treats the other variable as constant, so the integration “constant” may depend on that other variable.

[L05 | Bernoulli] State Bernoulli’s equation for steady, incompressible, inviscid flow with no body forces.

p + ½ρV² = constant along a streamline.

[L05 | Bernoulli assumptions] List the assumptions for the lecture form of Bernoulli’s equation.

Steady, inviscid, incompressible flow with no body forces; the comparison is along a streamline unless the flow is irrotational.

[Exam | Bernoulli] When does one Bernoulli constant apply everywhere in the flow field?

When the flow is also irrotational.

[Exam | Bernoulli] True or false: Bernoulli holds everywhere in a steady, incompressible, inviscid, irrotational flow.

True.

[L05 | Venturi] In a steady incompressible duct, what happens to speed as area decreases?

Speed increases because AV is constant.

[L05 | Venturi] In a horizontal ideal duct, what happens to static pressure where speed increases?

Static pressure decreases by Bernoulli’s equation.

[Exam | Duct comparison] If A₁>A₂ in steady incompressible ideal flow, compare V₁ and V₂.

V₁ < V₂.

[Exam | Duct comparison] If A₁>A₂ in steady incompressible ideal flow, compare p₁ and p₂.

p₁ > p₂.

[Exam | Duct comparison] If A₁>A₂ in steady incompressible ideal flow, compare total pressures p₀,₁ and p₀,₂.

p₀,₁ = p₀,₂.

[L05 | Total pressure] Define total or stagnation pressure for incompressible flow.

p₀ = p + ½ρV².

[L05 | Pitot] What does the forward-facing Pitot opening measure?

Total/stagnation pressure because the flow is brought to V=0 at the opening.

[L05 | Pitot] What does a static port measure?

Static pressure.

[L05 | Cp] Define pressure coefficient.

Cₚ = (p-p∞)/q∞ = (p-p∞)/(½ρ∞V∞²).

[L05 | Cp velocity form] Under incompressible Bernoulli, express Cₚ in terms of local speed.

Cₚ = 1 - (V/V∞)².

[L06 | Potential flow assumptions] What two conditions produce Laplace’s equation for φ?

Incompressibility (∇·V=0) and irrotationality (V=∇φ).

[L06 | Laplace] State Laplace’s equation for velocity potential.

∇²φ = 0.

[L06 | Laplace 2D] What equation does ψ satisfy in 2D incompressible irrotational flow?

∇²ψ = 0.

[L06 | Laplace meaning] Is Laplace’s equation linear or nonlinear?

Linear (and second-order), so solutions can be superposed.

[L06 | Superposition] State the method of superposition for potential flows.

Any linear combination of solutions to Laplace’s equation is also a solution.

[L06 | Infinity BC] What is the infinity/freestream boundary condition for flow in the +x direction?

As distance→∞, u→V∞ and v→0; equivalently φ→V∞x.

[L06 | Wall BC] State the inviscid wall boundary condition.

No penetration: V·n = 0.

[L06 | Wall BC ψ] Express the wall condition using ψ.

The body surface is a streamline, so ψ = constant along it.

[L06 | Uniform flow] Give φ and ψ for uniform flow V∞ in the +x direction in Cartesian coordinates.

φ=V∞x and ψ=V∞y.

[L06 | Source/sink velocity] Give the velocity field for a 2D source/sink.

Vᵣ=Λ/(2πr), Vθ=0.

[L06 | Vortex velocity] Give the velocity field for the course vortex sign convention.

Vᵣ=0 and Vθ=-Γ/(2πr).

[L06 | Vortex sign] Under the course convention, what direction does Γ>0 rotate?

Clockwise, because Vθ is negative.

[Exam | Equipotential sketch] How should equipotential lines be sketched for a vortex?

As radial lines through the origin; direction arrows are not placed on equipotential lines.

[Exam | Equipotential sketch] How should equipotential lines be sketched for a sink?

As concentric circles centered at the sink; direction arrows are not placed on equipotential lines.

[L07 | Doublet definition] How is a 2D doublet formed?

Bring an equal-strength source and sink together while letting their separation ℓ→0 and keeping κ=Λℓ constant.

[L07 | Doublet ψ] Give the doublet stream function.

ψ=-(κ/2π)(sinθ/r).