Jacek Spychalski
Numerical Analysis for Engineers
(NAE C_S2001)
(MEAE-4960)
TERM PROJECT
Laminar Flow near a Rotating Disk
(H. Schlichting, “Boundary Layer Theory”, 7th edition, p.102)
Class Instructor:
Prof. Ernesto Gutierrez-Miravete
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CONTENTS:
Page:
INTRODUCTION
3
PHYSICAL DESCRIPTION OF THE PROBLEM
4
MATHEMATICAL FORMULATION OF THE PROBLEM
5
METHODOLOGY OF THE SOLUTION
10
NOMENCLATURE
14
REFERENCES
15
2
INTRODUCTION
This project was chosen to demonstrate applicability and accuracy of numerical
methods used in solving a problem of the Boundary-Layer Theory.
Modern
mathematical tools, supported by computers, allow interpreting, analyzing and computing
physical models with incredibly high precision. Boundary-Layer theory is one of the
most important theories of fluid mechanics. At first it was developed for theoretically
ideal incompressible fluids in the conditions of laminar flow. Due to extended research
through many years, scientists were able to expand the existing theory and find an
application for many other non-incompressible fluids. Prandtl was the first who
introduced the theory in 1904. He was able to present the solution neglecting viscosity
but, as later research showed, his theory did not fully explained practical experiments. J.
P. Hartnett and E. R. G. Eckert were among next who contributed into the BoundaryLayer Theory.
Their study concerned laminar boundary-layer equations for an
incompressible, two-component gas.
In this project problem of laminar flow near rotating disk was solved using the
Runge-Kutta-Fehlberg method.
In the result, a series of points were obtained and
numerical interpolation was used to obtain polynomial approximations of the functions.
Interpolated functions represented on the graph are similar to one depicted in the H.
Schlichting “Flow near a rotating disk”.
3
PHYSICAL DESCRIPTION OF THE PROBLEM
Physical model presented below consists of a rotating disk immersed in a large amount of
fluid. Motions within the fluid are generated by rotating disk, which induces mass
transfer phenomenon. Disk of a radius 'R' rotates around an axis perpendicular to the
surface with uniform angular velocity ‘w ’.
Due to the viscous forces n a layer of fluid
is carried by the disk. Mass transfer takes place at the surface of the disk where the layer
near the disk is being directed outboard by centrifugal forces. Fluid motion is
characterized by velocity components u-radial, v-circumferential, and w-axial. Fluid is
defined as single-component gas and therefore mass removal or addition is uniform at all
points on the disk surface.

4
MATHEMATICAL FORMULATION OF THE PROBLEM
Considering system of cylindrical co-ordinates:
2 
2 1 
2
1 2



r 2 r r z 2 r 2  2
The mathematical statement of mass conservation is expressed by four Navier-Stokes
equations.
u
  2u   u   2u 
u v 2
u
1 p

w

  2     2 
r r
z
 r
r  r  z 
 r
u
 2v   v   2v 
v uv
v
  w   2     2 
r r
z
r  r  z 
 r
  2 w 1 w  2 w 
w
w
1 p
u
w

  2 


r
z
 z
r r z 2 
 r
Although equations describe behavior of fluid at some distance from an object most
critical and important from physical point of view is laminar flow of the layer nearest to
surface of the disk z = 0. From physical and mathematical description we can determine
boundary conditions considering no-slip condition at the wall of the disk:
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z  0:
u  0, v  r * ω, w  0
z   : u  0, v  0
At first we need to define the thickness s of the fluid layer at the surface of rotating
disk, which is carried due to friction. That layer of fluid is spinning with equal angular
velocity w. Thickness of the layer depends on angular velocity and decreases when
disk accelerate. Since in our experiment angular velocity is constant, thickness of the
fluid layer resting on the surface of the disk will also remain unchanged. The centrifugal
force that acts on a fluid particle in the rotating layer at a distance r from the axis can be
presented as:
FC   * r *  2
Therefore for a volume of area centrifugal force becomes:
FC   * r *  2 *  * dr * ds
Due to nature of physic the same element of fluid interacts with shearing stress t that
points in direction in which the fluid is slipping. Angle Q is created between shearing
stress t and circumferential velocity v. It is understandable that the radial component of
the shearing stress must be equal with centrifugal force F.
6
 * sin  * drds   * r *  2 * drds
 * sin    * r *  2 * 
Gradient of circumferential velocity at the wall has to be proportional to circumferential
component of shear stress. That additional condition:
 * cos 
 * r *

allows us to eliminate  :
2 

tan 

Assuming that direction of slip near disk does not depend on the radius the thickness of
the fluid carried on the surface of the disk is:



After defining thickness of the fluid layer, which rotates with the disk at no-slip
condition, we need to analyze system of Navier-Stokes equations.
Solution for that system can be obtained much easier by transforming to reduce
partial differential equations into system of ordinary differential equations. Successful
attempt of solving similar velocity problem for an impermeable disk rotating in a singlecomponent fluid was achieved in 1921 by T. von Karman. In order to use similarity
7
transform to reduce the partial differential equations to ordinary differential equations
new variables need to be introduced:
Independent variable:
z


Dependent variables:
F ( ) 
u
r *
G ( ) 
P( ) 
v
r *
H ( ) 
w
 *
p
 * * 
Inserting these new variables into Navier-Stokes equations we transform partial
differential equations into system of three ordinary differential equations:
F "  HF ' F 2  G 2
G"  HG '2 FG
H '  2 F
with following boundary conditions:
F( 0 )  0, G( 0 )  1,
H( 0 )  0
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Two missing Boundary conditions:
F ' ( 0 )  0.510, G ' ( 0 )  0.6159
can be obtained by approximation method stretching of the independent variable and
using least squares method to minimize the error in the differential equations. This
approximate solution is presented in Journal of Transactions of the American Society of
Mechanical Engineers3, Vol. 63, June 1996. P. D. Ariel who was an author of that article
gives three boundary conditions of which two are essential for us: F’(0), G’(0).
Approximate solution still differs from exact one, but most accurate from other
approximating methods provided by Cochran (1934) and Jain (1963).
Exact
Cochran
Jain
Ariel
F’(0)
0.510233
0.54
0.506711
0.507415
-G’(0)
0.615722
0.54
0.626752
0.608686
-H(  )
0.884474
0.55
0.999596
0.913029
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METHODOLOGY OF THE SOLUTION
In order to use numerical method following system of three II-order equations:
F "  HF ' F 2  G 2
G"  HG '2 FG
H '  2 F
needs to be transformed in to the system of five I-order equations.
Two new dependable variables are introduced:
U  F'
 U'  F"
T  G'
 T'  G"
which gives:
U '  HU  F 2  G 2
T '  HT  2 FG
H '  2 F
F' U
G'  T
with boundary conditions:
F( 0 )  0, G( 0 )  1,
H( 0 )  0,
U (0)  0.510233,
T (0)  0.615722
Next, using Maple 6’s dsolve command we can imply Runge-Kutta-Fehlberg method to
obtain the approximate solution at   0, 0.2, 0.4, 0.6, 1,..., 3.8, 4.0
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Table 1.
F( x)
x
0.00000
0.20000
0.40000
0.60000
0.80000
1.00000
1.20000
1.40000
1.60000
1.80000
2.00000
2.20000
2.40000
2.60000
2.80000
3.00000
3.20000
3.40000
3.60000
3.80000
4.00000
0.000000000000000
0.083589480952195
0.136265813534278
0.165833552081648
0.178690530701615
0.179920392109042
0.173447233299229
0.162215225950845
0.148369146302710
0.133421380561185
0.118397897330523
0.103960435962273
0.090505243556561
0.078240485027101
0.067245315255996
0.057513838863340
0.048987033716891
0.041575359291862
0.035174334366611
0.029674928034580
0.024970206703253
G( x)
1.000000000000000
0.878017235759183
0.762094966677223
0.655721950584866
0.560447790195260
0.476584982214989
0.403700433571598
0.340950599956134
0.287302372441664
0.241672493472918
0.203010544370346
0.170344124950723
0.142799637086026
0.119608022645444
0.100101758851167
0.083707214442349
0.069934938366017
0.058369421609893
0.048659200937692
0.040507749786123
0.033665344188709
H( x)
0.000000000000000
-0.017893367818338
-0.062756077101281
-0.123831184747898
-0.193200689701784
-0.265238492632630
-0.336114288496107
-0.403365310828080
-0.465540724278065
-0.521915883438673
-0.572269337733240
-0.616713504957524
-0.655569481002495
-0.689276989070835
-0.718331518740803
-0.743241964654785
-0.764503344430223
-0.782580339307310
-0.797898403344166
-0.810840011898259
-0.821744275719483
Series of points presented above are the results of Runge-Kutta-Fehlberg
technique used in Maple. Independent variable x was chosen arbitrary and only some
values of the function: F, G, H in the domain [0, 4] is shown in the Table 1. More results
with increment 0.1 from x are in the Maple file (Laminar Flow.mws)
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After obtaining certain amount of data points one can apply numerical method to
demonstrate interpolating polynomials.
Also in this case Maple program was used to
generate interpolating polynomials. Three functions F, G, H were interpolating so that:
F(x) = f1(x), G(x) = f2(x), H(x) = f3(x)
f1 := .5002299938 x2.03799114432 x11.08867574000 x9.06523360822 x10
.04373395834 x6.07702094923 x7.09455164005 x8.2070579737 x3
.01504381964 x5.5100130996 x.1011778271 10-8 x20.8828014889 10-5 x17
.0004151882288 x15.7629542676 10-6 x18.006433937606 x13
.001854499745 x14.00007066208394 x16.4073339127 10-7 x19
.01756439078 x12.03950081198 x4
f2 := .00049121697 x2.08581949842 x11.1965668992 x9.1458259307 x10
.1195124231 x6.1745935248 x7.2090107999 x8.1737614287 x3
.09175682682 x5.2591645153 10-8 x20.00002162310060 x17
.0009877489501 x15.1896788587 10-5 x18.01489271994 x13
.004350620446 x14.0001705496384 x16.1027930292 10-6 x19.6158720540 x
.04014453318 x12.1264353633 x41
f3 := .5090927279 x2.00005187782 x.1489810186 x11.3451798560 x9
.2546044571 x10.1973773942 x6.3081078540 x7.3689798405 x8
.3264271601 x3.07917742357 x5.4350494083 10-8 x20.00003655974704 x17
.001681714246 x15.3198064845 10-5 x18.02558124266 x13
.007438321786 x14.0002892979369 x16.1729002934 10-6 x19
.06930925546 x12.07177697455 x4
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Maple’s plot command produced following graph:
x
where:
F(x)
G(x)
H(x)
Velocity components u, v, w are easy to obtain from previously introduced
equations:
F ( ) 
u
r *
H ( ) 
w
 *
G ( ) 
v
r *
Numerical methods used in this article approximated solution to required
accuracy and allow obtaining results within reasonable amount of time.
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NOMENCLATURE:
F, G, H
= dimensionless velocity variables
P
= dimensionless pressure
r
= radial coordinate
u, v, w
= velocity components
FC
= centrifugal force
r
= density
w
= angular velocity
d
= thickness of viscous layer
t
= shearing stress
n
= kinematic viscosity
x
= dimensionless independent coordinate
Q
= angle formed between shear stress vector and circumferential velocity
Z
= distance from the surface of the disk
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REFERENCES:
1.
H. Schlichting “ Laminar Flow near Rotating Disk”, Boundary Layer Theory, 7th ed.
p. 102.
2.
E. M. Sparrow, J. L. Gregg “Mass Transfer, Flow, and Heat Transfer About a Rotating
Disk” Journal of Transactions to ASME, November 1960, p. 294 – 302
3.
P. D. Ariel “The Flow Near a Rotating Disk: An Approximate Solution”, Journal of
Transactions of the ASME, June 1996, vol. 63, p. 436 – 438.
4.
R. L. Burden, J. D. Faires “Numerical Analysis”, 7th ed.
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