EAS 4101 Exam1 Spring2010 Solutions revised

EAS 4101 Exam1 Spring2010 Solutions revised - EAS 4101 –...

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Unformatted text preview: EAS 4101 – Aerodynamics – Spring 2010 02/17/10 Exam #1 Name: ________________________ I. TRUE/FALSE (10 Points total – 1 point each) 1. Extensive properties are dependent on the size of the system. 2. The Reynolds’ number, an important non-dimensional parameter in fluid mechanics, is the ratio of viscous to pressure forces. 3. Timelines, pathlines, streaklines, and streamlines are all identical in a steady flow. 4. Motion about the vertical axis of an airplane is called yaw, and is produced by deflection of the elevators. 5. The convective derivative of a flow property is written as · , where is the given flow property. 6. Flow in a boundary layer is irrotational in the presence of a favorable pressure gradient. 7. Assuming inviscid flow transforms the Navier-Stokes Equations into Euler’s Equations. 8. The substantial derivative is always equal to zero if the flow is steady. 9. Pathlines are a locus of particles that have passed through a given point. 10.Stokes Theorem states that the circulation of a vector around a curved surface is equal to the flux of the curl of that vector over a surface. II. T F F F T F T F F T SHORT ANSWER (30 points total – 4 pts each unless otherwise marked) 11. (2 pts) The velocity boundary condition at a solid surface for an inviscid-flow problem is described by: a. Normal velocity is equal to zero. b. No-slip condition holds. c. Tangential velocity is equal to the free stream velocity. 12. (2 pts) From an ideal flow solution, it is possible to integrate the pressure on the surface of a body to determine ____Pressure________ drag but not _____Viscous______ drag. 13. In ideal flow, it is common to employ superposition solutions of various individual solutions to Laplace’s equations in terms of the stream function and/or velocity potential. Is it also valid to superimpose pressure fields of each individual solution? Please justify your answer mathematically and/or physically. No, the pressure fields are non-linearly related to the velocity fields via Bernoulli’s equation in ideal flow. Name: ________________________ 14. Once the velocity potential is known for ideal flow over a body, how do you determine the lift? Describe the specific steps. 1. Compute velocity field according to 2. Using Bernoulli’s equation, determine the pressure distribution. 3. Integrate the pressure distribution over the area to determine the net force. 4 .Take the component of the force in the direction opposite gravity to determine the lift. 15. The pressure distribution over the surface of a rotating cylinder of radius R is often given . Write an integral expression for the lift per unit span as a pressure coefficient, acting on the cylinder in terms of . 1 2 1 2 sin sin 16. For a low-speed, steady flow around the thin airfoil shown below, the free stream pressure and velocity ( and ) are known. The local static pressure at points 2-6 are known from measurements. At which point(s) out of 2-6 may we use Bernoulli's equation to determine the local velocity? Circle your answer(s). In a single sentence, explain your choices. 2 3 4 5 6 Bernoulli’s equation is only valid when the flow is incompressible, irrotational, and inviscid; flow within a boundary layer is both rotational and viscous. 17. (6 pts) Starting with the integral form of the conservation of mass equation, obtain the substantial derivative form of continuity. (Hint: use Gauss’ theorem to convert the surface integral into a volume integral.) · 0 Assuming fixed control volume: · 0 Applying Gauss’ theorem: 0 · · From this it can be seen that the integrand must be equal to zero: · 0 · · · · , 0 · 1 · Name: ________________________ Problem 1: (20 points total) Given: A thin symmetric airfoil section is mounted in a low-speed wind tunnel. A Pitot probe is used to determine the velocity profile in the viscous region downstream of the airfoil. This velocity distribution is 1 1 cos , for 2 2 is a constant. The fluid is air, with a density . where The following integrals may come in handy: 1 1 sin 2 9 16 1 1 1 cos 2 9 16 1 Find: a) List appropriate assumptions. (3 pts) b) Sketch the proper control volume for this problem, clearly labeling the forces acting on the control volume and the appropriate unit normal vectors. (3 pts) c) Determine the relationship between and . (7 pts) d) Determine the drag force per unit span acting on the airfoil. Explicitly indicate its direction with an arrow. (7 pts) Name: ________________________ Problem 1continued: a) Required: Incompressible, steady. Allowed: Neglect changes in height, no flow across the streamline b) Start with the integral form of the conservation of mass: · 0 Assuming steady, incompressible flow, and noting that there is only flux across the left and right boundaries, · · 0 Dividing by , plugging in the velocity, and making use of the symmetry about 2 2 1 1 cos 2 2 0, 0 Evaluating the integral, 2 and solving for 1 2 0 , 1 c) Start with the integral form of the conservation of momentum: · . Assuming steady, incompressible flow, plugging in velocity, and integrating via the given integral formula with the problem, 2 2 1 1 cos 2 2 4 2 4 2 4 2 9 16 1 cos 2 1 9 16 1 1 in the negative x‐direction The force is the force required to hold the airfoil stationary and acts in the negative xdirection. Drag, by definition, is the force of the fluid acting on the airfoil, and thus is equal to and acting in the opposite direction. in the positive x‐direction 2 1 2 or 4 or 2 2 1 9 1 16 or 1 4 8 1 . 1 Name: ________________________ Problem 2: (20 points total) Given: Starting with the Navier-Stokes equation in Gibbs notation: derive Bernoulli’s equation. Hint: Use the following vector identity: · 1 2 Find: a) List appropriate assumptions. (5 pts) b) Derive Bernoulli’s equation. Show and explain all steps. (15 pts) Name: ________________________ Problem 2 continued: a) Incompressible flow / constant thermodynamic properties, Stokes hypothesis; Steady; Inviscid flow; Irrotational flow b) Applying assumptions to the N-S equation we obtain DV p f b 2V Dt (2) V (3) V V p f b 2V t Then V V p fb Recall the vector identity V V 1 V 2 2 V V Since flow is irrotational this equation reduces to 1 V V V 2 2 Plugging back into the reduced N-S equation we obtain 1 V 2 p fb 2 or 1 V 2 p fb 2 Where fb is the gravity force acting on the fluid: fb gz Then we obtain 1 V 2 p gz 2 or 1 1 V 2 p gz 0 2 After integrating, Bernoulli’s equation is 121 V p gz const 2 Name: ________________________ Problem 3: (20 points total) Given: , , Add a source at the origin r r θ Fig. a θ Fig. b Flow impinging on a flat plate (Fig. a) can be described with the stream function 1 Ψ sin cos sin 2 2 where is a constant. By adding a source at the origin , flow against a plate with a "bump" is obtained (Fig. b). In both figures, the streamline patterns are shown. The fluid is air. The following stream functions may be useful: cos Λ Λ Γ Ψ , Ψ , Ψ , Ψ ln 2 2 2 2 The velocity field is related to the stream function by: 1Ψ Ψ , Find: a) Use superposition to determine the stream function for the flow of Fig b. (2 pts) , , ̂ , ̂ for an arbitrary point within the flow. b) Find the velocity (2 pts) c) Find the bump height in terms of and the source strength. (4 pts) d) Identify the value of the streamline that describes the surface of the “bump.” (4 pts) e) Find an expression for the shape of the bump, . Write your answer in terms of and . (4 pts) , neglecting gravitational f) Find the gage pressure on the surface of the bump at effects. (4 pts) Name: ________________________ Problem 3 continued: a) 1 2 Ψ Λ 2 sin 2 b) 1Ψ Λ 2 cos 2 Ψ sin 2 c) Recognizing that the peak of the “bump” is a stagnation point: ⁄2, 0. At Then evaluating, , , Λ 2 0 2 Λ 2 d) Plugging in the location of the stagnation point Ψ Λ 4 e) First, use (d) to relate the two coordinates: Λ1 sin 2 42 and making use of (c) Λ 2 2 2 2 sin 2 f) First, evaluate at : 4 2 sin 2 2 To find the gage pressure, apply Bernoulli’s equation between a point in the free stream and the point where 1 2 1 2 Neglecting gravitational effects: 1 2 We know that: To find these values, we evaluate and Λ , 24 , at the point Λ , Λ 2 1 2 2 2 Λ 1 2 2 , 24 2 Thus Λ 1 2 Λ 2 1 2 2 2 Λ 4 2 2 2 Plugging this back into the expression for gage pressure yields (where any substitution of the above is sufficient): 1 2 ...
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