assignment 6 with solutions[2] fm white

7/25/2019 Assignment 6 With Solutions[2] Fm white

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MECH 3492 Fluid Mechanics and Applications Univ. of Manitoba Winter Term, 2015

Assignment 6

Please google hotwire velocimetry and wind-tunnel for information on their structures and func-

tions.

Problem 1 (Purpose: understanding Reynolds averaging, viscous and turbulent shear stresses, andeddy viscosity)

In a wind-tunnel experiment, hotwire velocimetry was used for measuring a turbulent wind field. The

recorded time series of instantaneous velocity data for u, v and w (in m/s) at two close points are

given in the excel book Turbulence.xlsx. The measurements were taken at 1000 Hz. The spatial

distances between the two points are x= x2 x1 = 2.12 mm, y = y2 y1 = 3.91 mm and z =

z2 z1= 1.88 mm. The density and dynamic viscosity of the air are 1.25 kg/m3 and 1.5 105 Pa s,

respectively.

(1) Determine time-averaged velocities u, v and w at both points;

(2) Plot uand uw.r.t. time in the same diagram for point 1;

(3) Using the measured data to verify the assumption of incompressibility (for the mean flow) at the

midpoint (between the two points);(4) Determine the value of the viscous shear stress tensor based on Stokess hypothesis at the midpoint;

(5) Determine the value of the Reynolds stress tensor at the midpoint;

(6) Determine the value of the eddy viscosity at the midpoint based on Boussinesqs assumption.

Hints:

(1) you would need to pay attention to the units;

(2) this represents a much simplified hotwire experiment, as only two measurement points are involved.

When you calculate the derivatives, use the first-order accurate scheme (e.g., du/dx u/x).

Problem 2(Purpose: understanding BL thickness, displacement thickness, momentum thickness, and

integral equation)

For a zero-pressure-gradient laminar BL flow over a flat plate, assume that the velocity profile is of a

polynomial form: u= a+by+cy3. Derive all the equations and parameters listed in row 3 of Table 9.2.

Specifically, determine: (1) uU

; (2) w; (3) non-dimensionalized BL thickness x

(as a function ofRex);

(4) Cf(as a function ofRex); (5)

; (6)

; and (7) shape factor H=

.

Problem 3(Purpose: understanding turbulent BL and wind-tunnel design)

9.21

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Problem 9.21 [Difficulty: 2]

Given: Data on wind tunnel and boundary layers

Find: Displacement thickness at exit; Percent change in uniform velocity through test section

Solution

:The solution involves using mass conservation in the inviscid core, allowing for the fact that as the boundary

layer grows it reduces the size of the core. One approach would be to integrate the 1/7 law velocity profile tocompute the mass flow in the boundary layer; an easier approach is to simply use the displacement thickness!

Basic

equations

(4.12) disp

0

y1u

U

d

Assumptions: 1) Steady flow 2) Incompressible 3) No friction outside boundary layer 4) Flow along streamline 5) Horizontal

For this flow U A const andu

U

y

17

The design data is Udesign 160ft

s w 1 ft h 1 ft Adesign w h Adesign 1 ft

2

The volume flow rate is Q Udesign Adesign Q 160ft

3

s

We also have in 0.4 in exit 1 in

Hence disp

0

y1u

U

d

0

y1y

1

7

d 0

1

1

1

7

d where y

disp

8

Hence at the inlet and exit

dispin

in

8 dispin 0.05 in dispexit

exit

8 dispexit 0.125 in


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Hence the areas are Ain w 2 dispin h 2 dispin Ain 0.9834 ft2

Aexit w 2 dispexit h 2 dispexit Aexit 0.9588 ft2

Applying mass conservation between "design" conditions and the inlet

Udesign

Adesign

Uin

Ain

0

or Uin Udesign

Adesign

Ain

Uin 162.7ft

s

Also Uexit Udesign

Adesign

Aexit

Uexit 166.9ft

s

The percent change in uniform velocity is thenUexit Uin

Uin

2.57 % The exit displacement thickness is dispexit 0.125 in

assignment 6 with solutions[2] fm white

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