sudden accelerated plate

SUDDEN ACCELERATED PLATE

BY:ANDI FIRDAUS SUDARMA

ID. NO.: 432107963

LECTURER:PROF. DR. ALI A. AL-SEIF

SEMESTER PROJECTADVANCE FLUID MECHANICS I (ME 583)

MAGISTER PROGRAM OF MECHANICAL ENGINEERINGCOLLEGE OF ENGINEERING

KING SAUD UNIVERSITYRIYADH, KSA.

SECOND SEMESTER 1432/1433 H

Semester ProjectAdvance Fluid Mechanics I (ME 583)

APPROVED FOR THE ADVANCE FLUID MECHANICSSEMESTER PROJECT

Prof. DR. Ali A. Al-Seif, Lecturer. Date: 23 Jumada II 1433 H14 May 2012 G

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ABSTRACT

This project investigated the behavior of velocity for flow near sudden accelerated flat plate. The velocity profile is obtained using similarity method (Stokes’ first problem) which gives solution in form of single ordinary differential equation. The obtained result is solved numerically using Simpsons’ approximation. The velocity and shear stress at the wall in varied time are also investigated.

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TABLE OF CONTENT

Abstract......................................................................................................................................3

Table of Content.........................................................................................................................4

List of Figures............................................................................................................................5

List of Tables..............................................................................................................................6

I. Introduction............................................................................................................................7

II. Theory...................................................................................................................................9

III. Result.................................................................................................................................13

IV. Discussion and Conclusions.............................................................................................17

References................................................................................................................................18

Appendix..................................................................................................................................19

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LIST OF FIGURES

Figure 1. Viscous flow of a fluid near a wall suddenly set in motion........................................7

Figure 2. Sketch for Stokes’ first problem; sudden accelerated plate........................................8

Figure 3. Velocity distribution above a suddenly accelerated plate.........................................12

Figure 4. Velocity profile near a suddenly accelerated plate for varied time..........................13

Figure 5. Shear stress at the wall for varied time.....................................................................15

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LIST OF TABLES

Table 1. The result of velocity profile as shown on Figure 3..................................................20

Table 2. The result of velocity profile versus boundary thickness for varied time..................20

Table 3. The result of shear stress at the wall compare to time...............................................21

Table 4. Properties of air at atmospheric pressure...................................................................21

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I. INTRODUCTION

One of the unsteady viscous flow problems is sudden accelerated plate. Consider the plate is stationary at t = 0, and then at t > 0 the plate begin to move in its own plane (x direction) with a constant velocity U 0 which is illustrated in Figure 1. This phenomenon can be found on the surface of aircraft wings when it starts to take off.

t < 0, fluid at rest

t = 0, wall set in motion

t > 0, fluid in unsteady flow

Figure 1. Viscous flow of a fluid near a wall suddenly set in motion. [2]

Solving this problem using separation variable method gives incorrect solution. Since the solution contains trigonometric function (sinus or cosines), while at certain value the solution gives negative value. In this case, velocity should never give negative value.

The other method is similarity formulation, which were solved by G. Stokes. The problem also known as Stokes’ first problem.

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II. THEORY

A semi-infinite body of liquid with constant and is bounded on one side by a

flat surface. Initially, the fluid and the solid surface are at stationary; but at time t = 0 the solid surface is set in motion on the positive x-direction

with a velocity . It is desired to know the velocity as a function of y and t. There is no pressure gradient or gravity force in the x-direction and the flow is assumed to be laminar.

Figure 2. Sketch for Stokes’ first problem; sudden accelerated plate.

In the first we examine the flow of an infinite body of fluid near a wall suddenly set in motion. This problem will illustrate the use of the similarity method, which enables one to reduce the partial differential equation to a single ordinary differential equation

We begin with Navier-Stokes’ equation for 2-D (x-y coordinate) incompressible flow; [4]

(Continuity equation) (1)

(Momentum eq. in x-direction)

(2)

(Momentum eq. in y-direction)

(3)

For sudden accelerated flat plate flow, we can consider that velocity in y direction (v) is negligible, while the flow is fully developed and pressure is constant. So, we can simplify the equation above as; [6]

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(4)

The initial and boundary conditions are; [2]

Initial Condition at

, for all (5)

Boundary Condition 1 at

, for all (6)

Boundary Condition 2 at

, for all (7)

Assume that the solution in form of;

(8)

Where (9)

and;

= function of t (time) only

= scaling function of t

= non dimensional velocity in form of function (non-dimensional coordinate)

In order to solve equation 4, each term must be defined. For the left hand side term can be written as

Where

(10)

Applying boundary condition 1 (equation 6), and , to equation 4 will simplify the equation into

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(11)

Differentiating equation above with respect to t will give

(12)

And integrating equation 11 w.r.t. t will obtain

(13)

Substituting equation 11, 12 and 13 into equation 4,

(14)

Let

, then obtain g(t)

integrating both side w.r.t. t

where C = 0

then

(15)

Let a = 2 and substitute into equation 14, we obtain the following ordinary differential equation for (η) ,

(16)

With the boundary conditions f (0 )=1 and f ( ∞ )=0.

To solve the equation above, let f ' ( η )=S and f ' ' (η )=S '. Then we obtain

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integrating both side w.r.t. η

where C = 0

integrating w.r.t. η

Then we get

(17)

Were we have arbitrary selected η = 0 for the lower limit of the indefinite integral, which we cannot evaluate in closed form; changing the lower limit η = 0 to a different limit would simply change the value of constant C2, still undetermined. By applying boundary conditions then evaluate equation 17 to obtain C1 and C2,

Rewrite equation 17,

(18)

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III. RESULT

3.1. Velocity Profile

To obtain values of velocity f ( η ), we integrate equation 18 numerically using Simpsons’ approximation. [3]

For the integration in form of ∫a

b

f ( x )dx, can be solved by the following

formula

(19)

WhereΔ x=(b−a)/n, and n is an even arbitrary number.

Let set n equal to 10, and we obtainΔ x=η/10. We assume that value η (maximum limit of integral in equation 18) is lay between 0 and 2. So, Δ x=0.2.

The obtained values of f ( η ) are available on the appendix. The velocity distribution is represented in Figure 3, and it may be noted that the velocity profiles for varying times are ‘similar’, i.e., they can be reduced to the same curve by changing the scale along the axis of ordinates (see Figure 4).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

12


Figure 3. Velocity distribution above a suddenly accelerated plate.

3.2. Velocity Profile for Varies Time

We begin with equation 9, and then substitute into equation 15,

Then we obtain,

(20)

Assume that the surrounding fluid is air at temperature 200 C with atmospheric pressure. The properties of the corresponding fluid are available on the appendix. [4]

For time are varied, where t = 1, 5, 10 and 20, the velocity distribution is represented in Figure 4.

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t=1t=5t=10t=20

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Figure 4. Velocity profile near a suddenly accelerated plate for varied time.

3.3. Shear Stress at The Wall

To obtain the shear stress at the wall, use the following equation,

(21)

We begin with equation 19 and then substitute into equation 13.

(22)

The derivative of equation above w.r.t. y is

(23)

Substituting equation above into equation 21, while g (t )=2√υt

In whichυ=μ/ ρ. Simplify the equation above we get,

(24)

For time are varied, where t = 1, 5, 10 and 20 and is constant, the shear stress at wall is represented in Figure 5.

Next step is checking the dimension, where equation 24 should be inN /m2.

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0 2 4 6 8 10 12 14 16 18 200

0.0005

0.001

0.0015

0.002

0.0025

0.003

Figure 5. Shear stress at the wall for varied time.

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IV. DISCUSSION AND CONCLUSIONS.

The velocity profile on Figure 3 shown that at plate surface (η = 0), the flow and plate has the same velocity and for the flow that far away from surface (η = ∞) the velocity is zero. Considering that the flow has non slip condition and boundary condition give the limitation which is at t > 0 the velocity of flow on the surface (y = 0) equal to speed of plate and at y = ∞ the velocity is zero. So, we can conclude that Figure 3 and boundary conditions are gives the same velocity profile.

On Figure 4, we are comparing the velocity profile in variation time. Take into consideration that y is boundary layer thickness (shown as δ in Figure 2). As time is increased, the boundary layer thickness becomes thicker. It may be noted that the velocity profiles for varying times are ‘similar’, i.e., they can be reduced to the same curve by changing the scale along the axis of ordinates.

The shear stress at the wall for varied time comparison as shown in Figure 5, shows that at time where plate initiate to move (t = 0), the shear stress reach the maximum value. As time is increasing, the shear stress at the wall becomes smaller.

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REFERENCES

[1] Al-Seif, A. A., “Advance Fluid Mechanics I (ME 583); Lecture Notes” Mechanical Department, King Saud University, Riyadh, Second Semester 2012.

[2] Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena”, John Wiley & Sons, New York, 1960.

[3] Edwards, C. H., Penney, D. E., “Calculus”, 6th ed., Prentice Hall, New Jersey, 2002.

[4] Fox, R. W., McDonald, A. T., Pritchard, P. J., “Introduction to Fluid Mechanics”, 6th ed., John Wiley & Sons, New York, 2003.

[5] Hirasaki, G. J., “Transport Phenomena (CHBE 501); Lecture Notes”, Rice University, Texas, Fall 2007.

[6] Schlichting, H., “Boundary Layer Theory”, 7th ed., McGraw-Hill, New York, 1979.

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APPENDIX

Table 1. The result of velocity profile as shown on Figure 3.

η Sn f (η)

0 0 1

0.2 0.197365 0.777297

0.4 0.379653 0.571608

0.6 0.535154 0.396144

0.8 0.657671 0.257898

1 0.746825 0.157298

1.2 0.806745 0.089686

1.4 0.843939 0.047717

1.6 0.865262 0.023656

1.8 0.876553 0.010916

2 0.882075 0.004685

Table 2. The result of velocity profile versus boundary thickness for varied time

t = 1 t = 5 t = 10 t = 20

y f (η) y f (η) y f (η) y f (η)

0.00000 1.00000 0.00000 1.00000 0.00000 1.00000 0.00000 1.00000

0.00155 0.77730 0.00346 0.77730 0.00490 0.77730 0.00693 0.77730

0.00310 0.57161 0.00693 0.57161 0.00980 0.57161 0.01386 0.57161

0.00465 0.39614 0.01039 0.39614 0.01470 0.39614 0.02078 0.39614

0.00620 0.25790 0.01386 0.25790 0.01960 0.25790 0.02771 0.25790

0.00775 0.15730 0.01732 0.15730 0.02449 0.15730 0.03464 0.15730

0.00930 0.08969 0.02078 0.08969 0.02939 0.08969 0.04157 0.08969

0.01084 0.04772 0.02425 0.04772 0.03429 0.04772 0.04850 0.04772

0.01239 0.02366 0.02771 0.02366 0.03919 0.02366 0.05543 0.02366

0.01394 0.01092 0.03118 0.01092 0.04409 0.01092 0.06235 0.01092

0.01549 0.00469 0.03464 0.00469 0.04899 0.00469 0.06928 0.00469

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Table 3. The result of shear stress at the wall compare to time.

µ υ t τ w

0.0000181 0.000015 1 0.002637

0.0000181 0.000015 5 0.001179

0.0000181 0.000015 10 0.000834

0.0000181 0.000015 20 0.00059

Table 4. Properties of air at atmospheric pressure. [4]

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