Old Finals

MECH305 Fluid Mechanics

Fall 1998

INSTRUCTOR: Dr. Ajay K. Prasad, 228 SPL

1994

A large open tank, shown below, contains water at a height H. The water is allowed to flow out through a pipe whose cross-sectional area is A. The pipe has a nozzle at the end, whose area is a. Assuming frictionless flow, determine (i) the height h to which the water rises in the piezometer tube, and (ii) the velocity V at the nozzle, for the following three cases:
a = 0, i.e., the nozzle is blocked and no flow is allowed.
a = A/2
a = A

Answer: (i) h = H; (ii) h = 0.75H; (iii) h = 0
Experiments show that the fluid velocity u very near a wall in turbulent flow varies only with distance y from the wall, wall shear stress, tau_w, and the fluid properties rho and mu.
You are required to form the dimensional matrix, find its rank, determine the correct number of Pi's, determine those Pi's, and then rewrite the relation u = f(y, tau_w, rho, mu) in dimensionless form.

Answer: u/sqrt(tau_w/rho) = F(u y rho /mu)
A 2 m wide water tunnel has an object mounted at the center for drag measurement, as shown below. Far upstream, the velocity profile is uniform and the velocity is 2 m/sec. After passing over the object, the velocity profile gets rearranged as shown, such that the velocity now varies linearly across the channel width.
(i) Determine the expression for the exit velocity, u_1 = f(y).
(ii) Next, determine the drag force (per unit depth into the page) on the object, given that the pressure difference (P_0 - P_1) = 3 kPa. You may neglect friction between the walls and the water.
Hint: Since the problem is symmetric about the center line, you may work with only the upper half of the channel width (0 < y < 1 m).

Answer: u_1 = 4y; Drag on object = 3.33 kN/m (acting to right)
A prototype ship is 36 m long and designed to cruise at 12 m/s (about 23 knots). Its drag is to be simulated by a 1 m long model pulled in a tow tank. For Froude number scaling, find (i) the tow speed, and (ii) the ratio of prototype to model drag.
Answer: V_m = 2 m/s; Drag ratio = 36^3
A certain two-dimensional, incompressible shear flow near a wall, as shown below, has the velocity component
u = U ( 2y/ax - y^2/a^2 x^2)

where a is a constant. Derive, from continuity the velocity component v(x,y) assuming that v=0 at the wall (y=0).
Answer: v = U (y^2/(a x^2) - 2 y^3/(3 a^2 x^3) )

1995

The drag on a 2-m diameter satellite dish due to an 80 km/hr wind is to be determined through a wind tunnel test using a geometrically similar 0.4-m diameter model dish. Assume standard air for both model and prototype. At this velocity, air is essentially incompressible.
At what air speed should the model test be run?
With all similarity conditions satisfied, the measured drag on the model was determined to be 170 N. What is the predicted drag on the prototype dish?
Answer: V_m = 400 km/hr; D_m = D_p = 170 N
Assume that the flowrate Q of a gas from a smokestack is a function of the density of the ambient air, rho_a, the density of the gas, rho_g, the acceleration due to gravity, g, and the height and diameter of the stack, h and d, respectively.
How many Pi's exist for this problem?
Determine the Pi's.
Answer: Q/sqrt(g d^5) = F(h/d, rho_g/rho_a)
Air discharges from a 5 cm diameter nozzle and strikes a fixed curved vane as shown in the figure. A stagnation pressure probe connected to a water U-tube manometer is located in the jet. The U-tube records a displacement of 20 cm. Neglect weight of the air, and all friction. Assume that the probe produces a negligible disturbance on the air flow. (rho_a = 1.2 kg/m^3; rho_w = 1000 kg/m^3.
Determine the horizontal component of the force exerted by the jet on the vane.
Determine the vertical component of the force exerted by the jet on the vane.

Answer: Horizontal force = 14.37 N (acting to right)
Vertical force = 3.85 N (acting up)
If viscous (frictional) effects are neglected, and the tank is large, determine the flow rate from the tank shown in the figure.

Answer: Q = 0.0133 m^3/s
The u velocity component of an incompressible, two-dimensional velocity field is given by:
u = 2xy
You are told that the flow is two-directional (only u and v are non-zero). You are also told (i) that the flow satisfies continuity, and (ii) that it is irrotational (i.e., vorticity is zero).
Determine v as a function of x and y.
Answer: v = x^2 - y^2 + const.

1996

A 1/5 scale model automobile is tested in a wind-tunnel with the same air properties as the prototype. The prototype velocity is 50 km/hr. For dynamically similar conditions the model drag is 350 N.
Determine the drag force on the prototype.
Determine the power in kilowatts required to overcome the drag force on the prototype.
Answer: Drag = 350 N; Power = 4.86 kW
Two equal solid, elastic spheres of radius R are pressed together by a force F as shown. The spheres deform to produce a circular contact region with radius r.
Using r, R, F and E (elastic modulus) form the dimensional matrix. What is the rank of this matrix? (rank = size of largest non-zero determinant).
How many Pi's exist in this system? What are they?
It is known that r varies as F^{1/3}. How then does r vary with R?

Answer: Two Pi's; r/R = F(F/ER^2); r ~ R^{1/3}
A two-dimensional, steady, flow field of constant density rho has velocity components u = Cx, and v = -Cy. Using the Navier-Stokes equations, derive the expression for pressure, p(x,y). Because the flow is 2-d, w=0 and all variations in the z-direction are zero. Neglect gravity.
Answer: p = - rho C^2/2 ( x^2 + y^2) + const.
A dredger is loading sand (specific gravity = 2.65) onto a moored barge as shown in the figure. The sand leaves the dredger pipe at 2 m/s, with a mass flux of 400 kg/s. What is the tension on the mooring line caused by this sand loading?

Answer: F = 692 N (acting to left)
A water jet 9 cm in diameter and Q = 20,000 cm^3/s flows over a fixed cone of half-angle theta = 35 deg and base diameter 40 cm as shown. If the water leaves as a conical sheet with the same velocity as the jet,
What is the force F in newtons required to hold the cone fixed?
How does this force vary with the base diameter?

Answer: F = 11.35 N acting to left; F is independent of base diameter D, it depends on theta only.

1997

A tundish is a shallow, open container used in the steel-making industry. The flow characteristics of molten steel in the tundish impacts the final quality of the steel product.
A transparent Plexiglas model is to be constructed to visualize the flow using water as the working fluid. Because this problem involves an open fluid surface, surface waves will be present; surface tension is also significant.
Write down the two non-dimensional groups that should be matched between model and prototype to ensure dynamic similarity.
What is the ratio, S, between the length scales of the prototype and the model?
Properties for water: rho = 1000 kg/m^3; gamma = 0.0073 N/m
Properties for molten steel: rho = 7080 kg/m^3; gamma = 1.6 N/m
Answer: Match Froude and Weber numbers.
S = 5.56
A weir is an obstruction in a channel flow which can be used to measure the mass flux mdot of the fluid. The mass flux is a function of the fluid density rho, gravity g, upstream fluid height above the water crest h, and weir width (into the paper) b.
From the given list of physical parameters, form the dimensional matrix and determine its rank. How many Pi's exist in this system?
It seems logical that the mass flux mdot is directly proportional to the width of the weir b. If so, how does mdot vary with h?

Answer: 2 Pi's: b/h and mdot^2/(rho^2 g h^5)
mdot ~ h^3/2
The cart shown below moves at a constant velocity V_o = 12 m/s and takes on water with a scoop 80 cm wide (into the paper) which dips h = 2.5 cm into a pond. Neglect air drag and wheel friction. Determine the force required to keep the cart moving.

Answer: 2880 N
A 2 m diameter fan is mounted inside a duct as shown. Air is drawn from the atmosphere and is discharged through a 1 m diameter exit at 40 m/s. Assume one-dimensional, incompressible, frictionless flow. The exit is open to the atmosphere. (For air, use rho = 1.2 kg/m^3; atmospheric pressure = 101.33 kPa.)
What are the pressures at sections 1 and 2?
What is the power added to the air by the fan?
What is the force required to hold the duct in place?

Answer: p_1 = -60 Pa; p_2 = 900 Pa;
Power = 30.16 kW
F = 1320 N (acting to right)
A fluid of viscosity mu is contained between two horizontal, infinite flat plates, separated by a distance h. The lower plate is fixed, while the upper plate moves to the right at a velocity U_o as shown.
Assume that the fluid is incompressible, the flow is one-directional, one-dimensional, and steady. The flow is driven purely by the motion of the upper plate, i.e. all pressure gradients are zero.
Apply the Navier-Stokes equations to the coordinate system shown in the figure. Reduce it to a simple ODE (provide reasons for each simplification!). Solve the ODE with appropriate boundary conditions.
Derive an expression for the velocity profile, u = f(y).
What is the shear stress across the fluid?

Answer: u = U_o/h * y;
shear stress = mu U_o/h