550 Lab 2 (Dimensional Analys)

Objectives: 1. To learn and practice basic principles of dimensional analysis 2. To apply dimensional analysis to description of some simple hydraulic problems Please put all answers on the attached answer sheets. Show all work and derivations.

LIST OF SYMBOLS

D diameter (ft, m) F any force (lb, N) γ specific weight (lb/ft3, N/m3) g gravitational (ft/s2, m/s2) ρ density (slug/ft3, kg/m3) acceleration h hydraulic head (ft, m) µ dynamic viscosity (lb-s/ft2, N-s/m2) L any length (ft, m) ν kinematic viscosity p fluid pressure (lb/ft2, N/m2) σ surface tension (lb/ft, N/m) Q discharge (ft3/s, m3/s) τ shear stress (lb/ft2, N/m2) r radius (ft, m) t time s U fluid velocity (ft/s, m/s) R Reynolds number dimensionless F Froude number dimensionless

I. BASIC PRACTICE IN DIMENSIONAL ANALYSIS Arrange each of the following groups of variables into dimensionless parameters:

a. τ, U, ρ

b. ∆p, U, γ, g

c. F, ρ, L, U

d. U, L, ρ, σ II. APPLICATION OF DIMENSIONAL ANALYSIS TO HYDRAULIC PROBLEMS

In each problem below, first write the basic functional relations (for example, FD = f(U,D,µ,ρ)), then write the relation in general terms (for example, FD = CUaDbµcρd), then substitute in the dimensions, and solve for the required relations and/or dimensionless numbers. Please show all your steps.

a. Derive an expression for the fluid drag FD on a submerged streamlined body. The parameters involved are: size of the body L; the velocity of the body U; viscosity of the water µ; and density of the water ρ. The size of the streamlined body may be represented by its diameter or its length. Now make the drag force dimensionless.

b. Derive a dimensionless expression for the drag force FD on a surface vessel. Use the same parameters as in problem a above, but add the acceleration of gravity g to account for wave action. Now make the drag force dimensionless. c. Derive a dimensionless expression for the flow rate q over the spillway shown in the sketch below. q is the flow rate per unit length of spillway perpendicular to the sketch (i.e., into the paper). Assume that the sheet of water is relatively thick, so that surface-tension effects may be neglected, and also assume that gravity effects predominate so strongly over viscosity that viscosity may be neglected. Relevant variables are: q = water discharge per unit length of spillway crest H = head of water above top of spillway Z = height of spillway

g = acceleration of gravity ∆

H q

Note: the left-hand side of the equation should be a dimensionless flow rate; the right-hand side should be an unspecified function f of another dimensionless quantity. d. Referring to the diagram above, derive a dimensionless expression for small flow rates q over the spillway, assuming that fluid density ρ and fluid viscosity µ cannot be neglected.

e. Derive a dimensionless expression for the shear stress τ at the wall of a pipe when an incompressible fluid flows in the pipe under pressure. Relevant partameters are flow velocity U, pipe diameter D, fluid viscosity µ, and fluid density ρ. Geology 550 Name: ______LAB 2 ANSWER SHEET I . Basic Practice. Show work

a. dimensionless number: ______

b. dimensionless number: ______

c. dimensionless number: ______

d. dimensionless number: ______II. Application to hydraulic analysis a. Basic functional relation:______

General equation: ______

Derived expression: ______

b. Basic functional relation:______

General equation: ______

Derived expression: ______c. Basic functional relation:______

General equation: ______

Derived expression: ______

d. Basic functional relation:______

General equation: ______

Derived expression: ______e. Basic functional relation:______

General equation: ______

Derived expression: ______