Table of Contents
ASSIGNMENT 1 (Due September 4, 2007)

[Original due date: August 30] — David Wagner 2007/09/03 12:02

Reading: Chapter 1

Questions: (Due August 30, 2007 )

Problems: Chapter 1

Prob 1, 2, 3, 4, 5, 6, 10, 13

Problem 1.

Steady flow occurs when

  • a. Conditions at all point of interest steadily change with time
  • b. Conditions are the same at adjacent points at any instant
  • ✔c. Conditions do not change with time at any point

Problem 2.

Turbulent flow occurs in situations involving

  • a. Very viscous fluids
  • b. Very small velocities of flow
  • c. Capillary tubes
  • ✔d. None of the above

Problem 3.

Eddy viscosity, defined in Eq. (1.3.36), is

  • a. A physical property of the fluid
  • b. The viscosity divided by the density of the fluid
  • ✔c. Dependent on the flow and density
  • d. Independent of the nature of the flow

Problem 4.

Viscous forces are weak relative to inertial forces in

  • a. Laminar flows
  • ✔b. Turbulent flows

Problem 5.

The Reynolds number may be defined as the ratio of

  • a. Viscous forces to inertial forces
  • b. Viscous forces to gravity forces
  • c. Gravity forces to inertial forces
  • d. Pressure forces to elastic forces
  • e. Inertial forces to gravity forces
  • ✔f. None of the above [ratio of inertial to viscous, Text §1.2.2]

Problem 6.

The Froude number may be defined as the ratio of

  • a. Viscous forces to inertial forces
  • b. Viscous forces to gravity forces
  • c. Gravity forces to inertial forces
  • d. Pressure forces to elastic forces
  • ✔e. Inertial forces to gravity forces [Text §1.2.2]
  • f. None of the above

Problem 10.

Given the schematic figure below of a constant width rectangular channel, answer the following questions.

Illustration for Problem 10

a. Estimate the net force on the sluice gate if all losses are neglected.

u_1 = 40/20 = 2; y_1 + {{{u_1}^2}/{2 g}} = y_2 + {{{u_2}^2}/{2 g}} = 20+{{2^2}/{2 g}} = y_2 + {{{q_2}^2}/{2 g {y_2}^2}} right {y_2}^3 - 20.06 {y_2}^2 +24.84=0

From Polynomial Web Solver: 

For the real Polynomial:
+1x^3-20.062x^2+24.84
The Solutions are:
X1=19.999899375114747
X2=-1.0838354981853493
X3=1.1459361230706027
Time used: 15 msec.

y_2 =1.15 ft; u_2=40/1.15 =34.8 ft/sec

F_1 = {gamma {y_1}^2}/2 = {62.4*20^2}/2=12480 lb/ft

F_2 = {gamma {y_2}^2}/2 = {62.4*1.15^2}/2=41.3 lb/ft

F_1 - F_2 - eq(u_2 - u_1) = 12480-41.3-1.94*40*(34.8-2)=9890

9890 lb/ft

b. Sketch the pressure distribution on the surface AB (note that the pressures at points A and B are zero gage pressure).

Answer to problem 10b.

c. Is the pressure distribution sketched in Part (b) a hydrostatic pressure distribution?

No

The pressure distribution is not triangular.

d. How is the pressure distribution sketched in Part (b) related to the force estimated in Part (a)?

The total force (per unit width) on the gate is the sum of the pressure distributed over the gate (area under/to the right of the curve).

F = int{A}{B}{P(y)}

Problem 13.

Water is flowing in a rectangular channel of unit as width shown in the figure below. Neglecting all losses, determine the possible depths of flow at Section B.

Illustration for Problem 13

Q_1 = Q_2 = u_1 A_1 = u_2 A_2 = u_1 d_1 w = u_2 d_2 w approx u_1 y_1 w = u_2 y_2 w <m>right u_2 = {u_1 y_1}/{y_2}

z_1 + {{{u_1}^2} / {2 g}} + y_1 = z_2 + {{{u_2}^2} / {2 g}} + y_2 = z_2 + {(u_1 y_1)^2}/{2 g {y_2}^2} + y_2 =8 + {{{16.1}^2} / {2*32.2}} + 4 = 0 + {(16.1*4)^2}/{2*32.2*{y_2}^2} + y_2 = 8+4.025+4={64.4/{{y_2}^2}}+y_2 = 16.025

right 64.4 + {y_2}^3 = 16.025{y_2}^2 right {y_2}^3 - 16.025{y_2}^2 + 64.4 = 0

From Polynomial Web Solver: 

For the real Polynomial:
+1x^3-16.025x^2+64.4
The Solutions are:
X1=15.765911770206875
X2=-1.895682748771265
X3=2.15477097856439
Time used: 16 msec.

Depths that make sense for this problem are the first and second roots listed: 15.8' and 2.2'.

15.8' and 2.2'

Discussion


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