me2135 12/13 tutorial solutions 1-10
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7/28/2019 ME2135 12/13 Tutorial Solutions 1-10
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ME2135 FLUID MECHANICS II – 2012/13
Part 2: Viscous Flow
Tutorial Solutions
1. (a) The flow is unsteadybecause time t appears in the velocity components.The flow is 3-D because all three velocity components are non-zero.
ka jaia
kdt
dw j
dt
dvi
dt
du
dt
Vda
zyx
where:
xtx
xzytttxx
z
uw
y
uv
x
uu
t
u
dt
du
Dt
Duax
2
2
164
0402444
ytty
xztyttxty
z
vw
y
vv
x
vu
t
v
dt
dv
Dt
Dvay
4
22
44
0422044
zxzxt
xxzytztx
z
ww
y
wv
x
wu
t
w
dt
dw
Dt
Dwaz
2
2
1616
4402440
kzxtxz jyttyixtxdt
Vda 242 161644164
At 0,1,1,, zyx :
k jttita 014414 32 Answer
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(b) Recalling that V = u i + v j + wk
where: zcybxau 111
zcybxav 222
zcybxaw 333
For incompressible flow:
0 V – continuity
or 0
z
w
y
v
x
u
Substituting u, vand w into the continuity will yield:
0321 cba Answer
(c) Again applying the continuity condition:
0 zw
yv
xu – for incompressible flow in Cartesian coordinates
Hence: 022
z
b
y
vayax
x
that is:
axy
v
y
vax 202
Integrating partially with respect to y gives:
tzxf axytzyxv ,,2,,, Answer
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(d) A 2-D velocity field:
jyxixyxV 222
xyxu 22and yxv 2
jdt
dvi
dt
du
dt
Vda
yx aa
yyxxxyx
y
uv
x
uu
t
u
Dt
Du
dt
duax
2212
22
yxxyx
y
vv
x
vu
t
v
Dt
Dv
dt
dvay
2222
At 2,1, yx : iax 10 and jay 8 Answer
For steady, 2-D flow, the rate of change of pressure is:
yyxxxyx
y
pv
x
pu
t
p
Dt
Dp
dt
dp
829 222
At 2,1, yx : units46Dt
Dp Answer
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sing
dzds
s
V
streamline
2. Bernoulli’s Equation is for ideal fluids, that is, viscosity ν = 0
For ν = 0, the Navier-Stokes Equations become:
1
1 1
1
Bx
B By
Bz
Du Pf
Dt x
DV Dv Pf P f Dt Dt y
Du Pf
Dt z
Apply Euler’s Equation along a streamline in a steady, incompressible plane
flow sVV , where s= unit vector.
sindsdz
ss
VV
t
V
s
VV
t
V
Dt
VD
0 (for steady flow)
in s-direction: s
P
gs
V
V
1
sin
s
P
s
zg
1
that is: 01
s
zg
s
P
s
VV
or: 02
2
gz
PV
s
or: gzPV
2
2
constant (Bernoulli’s Equation)
(Euler’s Equation)
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3.
Let the flow be parallel to the x-axis.
For a steady, 2-D, parallel flow (in Cartesian coordinate system):
2
2
y
u
dx
dP
Since 0xu (from Continuity equation, that is, u is independent of x):
2
2
2
2
dy
ud
y
u
dx
dP
dy
ud
2
2
with the following boundary conditions:
0u at by (no-slip)
0dy
duat 0y (symmetry of flow)
Integrating with respect to ygives:
Aydx
dP
dy
du
Applying the symmetry of flowboundary condition:
ydx
dP
dy
duA 0
Integrating again with respect to ygives:
By
dx
dPu
2
2
y
x
L1 2
u
U 2b=h
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Applying the no-slipboundary condition:
dx
dPbyu
b
dx
dPB 22
2
2
1
2
Noting that at 0
y , Uu
would yield: dx
dPbU
2
2
Comparing the above with the previous result, the following is obtained:
2
1
b
y
U
u Proven
The volume flow rate per unit depth of channel:
dx
dPbyb
y
dx
dPdyby
dx
dPdyuq
bbb
b
3
0
23
0
22
3
2
3
1
2
22
Since h = 2bor hb21 , then:
dx
dPhq
12
3
that is, a discharge in the positive x-direction requires a negative pressure gradient
or a pressure field that is decreasing in the flow direction.
Spatial average bulk velocity V :
dxdPh
hqV
12
2
2
1
22
112
dPV
hdx
PL
PPV
hxx
12
2
1212
NOTE: pressure drop 21 PPP 2
12
h
LVP
OR: h
L
hVhV
VL
V
P
24
12
22
212
21
But
hVRe , hence: h
L
V
P
Re
24
221
Proven
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NOTE: The use of the no-slip boundary condition ( 0u at by ) is sufficient to
obtain the velocity profile u.
From:dx
dP
dy
ud
2
2
Aydx
dP
dy
du
Hence: BAyydxdP
dydu 2
2
For 0u at by :
220
22 b
dx
dPAbBBAb
b
dx
dP
For 0u at by :
BAbb
dx
dP
20
2
Inserting the previous result:
220
22 b
dx
dPAbAb
b
dx
dP
2and020
2b
dx
dPBAAb
2222
2
1
22by
dx
dPb
dx
dPy
dx
dPu
The average velocity in the gap V is formulated as:
h
bdx
dP
h
yby
dx
dP
h
dyu
area
rateflowvolumeV
b
b
b
b
323
3
4
2
1
32
1
dx
dPh
dx
dP
h
bV
123
2 23
Integrating between x= 1 and x= 2 :
2
1
2
1
2
12dp
V
hdx
PL
PPV
hxx
12
2
1212
L
PhV
12
2
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4. (a) Plane Couette Flow: lower plate is stationary, upper plate moves to the right at velocity U
From: dx
dP
y
u
2
2
Aydx
dP
y
u
BAyy
dx
dPu
2
2
Boundary conditions:
0u at 0y : 0B
Uu at hy : Ahh
dx
dPU 2
2
2
12h
dx
dPU
hA
22
22 h
dx
dPU
h
yy
dx
dPu
h
y
h
yh
dx
dPy
h
U
yhydxdPy
hU
h
dx
dPU
h
yy
dx
dPu
2
22
2
22
2
1
2
1
2
1
2
1
h
y
h
y
dx
dPhy
h
Uu 1
2
2
Answer
Volume flow rate across the plates per unit z-width:
h
hh
x
hyy
dx
dPy
h
U
dyyhydxdPy
hUdyuq
0
232
0
2
0
232
1
2
2
1
dx
dPhUhqx
122
3
Answer
y
x
h
U
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4. (b) Plane Couette Flow: lower plate moves to the right at velocity U, upper plate is stationary
From: BAyy
dx
dPu
2
2
Boundary conditions:
Uu at 0y : BU
0u at hy : UAh
h
dx
dP
20
2
dx
dPhUhdyuq
hyy
dx
dP
h
yUu
h
x
122
21
3
0
Answer
y
x
h
U
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5.
2 212 2
2 212 2
0
u u u P u uu v f xt x y x x y
v v v P v vu v f yt x y y x y
u v
x y
// flow in the x-direction:2
20 0, 0
u uv
x x
For steady flow: ' 0st
Since 0 (given), the equation becomes:P
x
2
2
2
2
0 sin
sin
ugy
ug
y
With the boundary conditions: 1. at 0, 0y u
2. at , 0 0u u
y hy y
Integrating gives: sinu g y Ay
Applying the boundary condition 2: 0 sing y A
sinA gh
sin sinu
gy ghy
y
x
h
( )u y
g
sing
cosg
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Integrating again gives:
2
sin sin2
yu g ghy B
Applying the boundary condition 1: 0 B
Hence:
2
sin2
g yu hy
Answer
Flow rate per unit width Q is given by:
0
2
0
2 3
0
sin2
sin2 6
h
h
h
Q udy
g yhy dy
g y yh
3
sin .3
g hQ
Answer
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6. a) The assumptions made are:
(i) is constant
(ii) is constant
(iii) Two-dimensional (2-D) flow, since the equations of motion and
continuity do not include the azimuthal (angular) term
(iv) Body forces negligible
(v) Flow is steady no ' termst
For fully developed flow, the conditions are:
0 no flow in the r-direction
(parallel flow)
v
2
2also 0, 0
v v
x x
2
2and 0, 0
i.e. is independent of ,
hence:
u u
x x
u x
u du
r dr
Substituting the conditions to the equations of motions:
1 1
10, i.e.: is independent of
( ) only and
P d dur
x r dr dr
PP r
r
P dPP P x x dx
,r v
,x u
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b)
Fully developed velocity profile in the annulus of concentric cylinders:
where R 1 = radius of cylindrical cooling channel, and
R 2 = radius of fuel element (solid and co-axial with the cooling channel)
From the simplified equation:
2
2
1
1So that
2
1and
2
1ln
4
d du dPr r
dr dr dx
du dP rr A
dr dx
du dP r A
dr dx r
dP ru A r B
dx
with the boundary conditions:
2
1 1 1
2
2 2 2
1(1) 0 at 0 ln
4
1(2) 0 at 0 ln
4
dPu r R R A R B
dx
dPu r R R A R B
dx
2 2 11 2
2
1ln
4
dP RR R A
dx R
2 2
1 2
1
2
1
4
ln
dPR R
dxA
R
R
fuel
elementx
r
u(r)
R 1R 2
x
r
R 1
R 2
–
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2 2
1 2 12 2
1 1 1
1
2
1ln
1 14and ln
4 4ln
dPR R R
dP dPdxB A R R R
dx dxR
R
which gives:
2 2 2 2
1 2 1 2 12 2
1
1 1
2 2
ln ln1
4ln ln
R R r R R RdPu r R
dx R R
R R
Answer
is maximum when 0du
udr
2 2
1 2
1
2
1 12 0
4ln
R Rdu dPr
dr dx rR
R
2 2 2 2
1 2 1 22 2
1 1
2 2
2
ln 2ln
R R R Rr r
R R
R R
2 2
1 22 2
1 2 12 2
1 21 2 2
max 1
1 1
2 2
2 2
1 2
1
2
ln2ln
1
42ln ln
occurs at
2ln
R RR R RR RR RdP
u Rdx R R
R R
R Rr
R
R
Answer
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7 (a)
i)
Cαx bα
qμ6
αx bα
Uμ6x p
2
1
x
1
Eqn. (2.24) on Lecture Notes p. 23
with b.c.' s : 1) at x = 0, p = 0 (gauge) and 2) at x = L, p = 0 (gauge).
Substituting the b.c. 1) into Eqn. (2.24): Cα b
qμ6 bα
Uμ6 02
1
x
1
and the b.c. 2) into Eqn. (2.24): C
αL bα
qμ6
αL bα
Uμ6 0
2
1
x
1
From these two equations:αL) b(
U
b
U
αL)(b
q
b
q
11
2
1
x
2
1
x
whereL
b b α 21 Hence:
21
21x
21
2
2
x
2
1
x
b b
U b bq
b
U
b
U
b
q
b
q
ii) From Eqns. (2.25) and (2.22):21
21
21
21x
b b
b b2 b
2
bU
b b
U b bq
iii) b(x) = b1 – αx → xα b b whereL
b b α 21
21
1211
21
21
b b
L bx
L
x) b- b( b
b b
b b2
iv) The pressure distribution is given by Eqn. (2.26):
21
2
2
b bx b
x bx bUμ6x p
21
2
2mm
b b b
x b bUμ6 p b b(x)andxxat p p(x)
Substituting the expressions for b and x into the previous expression of pm ,
we will get:
2121
21m
b b
L
b b
) b b(Uμ
2
3 p
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7 (b) Given: 500 kN mF
w
Determine: i) of the lubricant
ii) Power/width
iii) Max. pressure pm
iv) Centre of pressure xcp
Solution:
i) From:
2
1 1 2
2
2 1 21 2
6ln 2
F UL b b b
w b b bb b
Equation (2.27)
Express in terms of the others and after substituting the corresponding
numerical values will give 20.0626 Ns/m 0.0626 Pa.s (or 0.626 cP)
ii)
1
2
ln2
F wD U b
w b
Equation (2.28)
where 1 2 0.075 0.025 0.050 0.001
150 150 3
b b
L
x1.5 m/sU
150 mmL
“toe” “heel”
b1
b2
b1 = 0.075 mm
b2 = 0.025 mm
centi Poise
(c.g.s. unit)
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3 0.0626 1.5500x10 0.001ln3
2 3 0.001 3
0.0626 4.55001.0986
6 0.001
83.333 309.476 N/m
392.81 N/m
D
w
Power consumed per width = x 589.213 W/mD w U
iii) From:
2
2
1 2
6 ( )( )
( )
U b x b xp x
b x b b
Equation (2.26)
maxp p occurs at x x and ( )b x b
where:
1 2
1 2
2(can you derive this?)
2 0.075 0.025= mm 0.0375 mm
0.075 0.025
bbb
b b
1 21
1 2
1
1 21 21
1 2
( ) where
2
b bb x b x
L
b b xb b
L
b b xbbb
b b L
Hence:
1
1 2
0.075 150mm 112.5 mm
0.075 0.025
bLx
b b
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Therefore,
2
max 2
1 2
2 3
3
6
6 0.0626 1.5 0.0125 112.5
0.0375 0.1x10
5634x10 Pa = 5.634 MPa
U b b xp
b b b
Be careful of the units !!
iv) Using:
1 1 2 1 1 2
2
2 1 21 2
1 2 1
1 2 2
2 5ln
2
ln 2
cp
b b b b L b bL
b b bb bx
b b b
b b b
Equation (2.33)
and substituting the numerical values:
1
2
0.075 mm (inlet gap)
0.025 mm (outlet gap)
150 mm (length of bearing)
b
b
L
will give xcp = 91.088 mm (that is, centre of pressure cp is 91.1 mm from the
“toe” or the inlet)
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8. Using volume flow rate per unit z-width in the x-direction:
h2
U
dx
dP
12μ
hq
3
x
(a) In zone 1 of the bearing: h = h1 andlmP
x p
Hence:2
h
P
12μ
hq 1m1
1
3
xoV
l
→ for zone 1
(b) In zone 2 of the bearing: h = h2 andlmP
x
p
Hence: 2
h
P
12μ
hq 2m2
2
3
x oV
l
→ for zone 2
(c) To satisfy continuity of flow :21
xx qq
Hence: 21m1m2 hh
2
P
12μ
hP
12μ
h
33
oV
ll
and therefore:
3321
21
hh
hhμ6
m
P
oVl
(d) Load per unit width F/w = Area of the pressure distribution
= ½ x 2 l x Pm = Pm l
(e) If h2 = ½ h1 Pm = 8 μ l Vo / 3 h12
and therefore: 21
2
22
2
h3
μ8
or h3
μ2
F/woo VlVl
If h2 = h1 F/w = 0 since there is no step, and hence Pm = 0.
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9. Velocity profile:n
1
δ
y
U
u
where n > 0
Boundary layer displacement thickness
δ
0
dyU
u1δ*
1n
δ
n
1)(n
δδ
1)n
1(
δ
δ
1δ
1)n
1(
y
δ
1ydy
δ
y1
1n
1
n
1
δ
0
1n
1
n
1
δ
0
n
1
Boundary layer momentum thickness
δ
0dyU
u1U
uθ
2)(n1)(n
nδ
2n
nδ
1n
nδ
1)
n
2(
δ
δ
1
1)
n
1(
δ
δ
1
1)
n
2(
y
δ
1
1)
n
1(
y
δ
1
dyδ
y
δ
ydy
δ
y1
δ
y
1n
2
n
2
1n
1
n
1
δ
0
1n
2
n
2
1n
1
n
1
δ
0
n
2
n
1δ
0
n
1
n
1
n
2n
θ
δH parameter shapeHence,
1Hn1δ
yyallfor Uu:layer boundary No
1.2857H;0972.09x8
7
δ
θ;
8
1
δ
δ7nFor (ii)
3H;6
1
3x2
1
δ
θ;
2
1
δ
δ1nFor (i)
n
1
NOTE: For a laminar boundary layer over a flat plate: H = 2.59
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10. For the velocity distribution:
yba
U
u
(a) To find a and b, use the boundary conditions:
0u at 0y (at the wall) 0a
Uu at y (at the edge of boundary layer) 1b
Hence, the velocity profile is
y
U
u or
yUu .
Boundary layer displacement thickness
0
1* dyU
u
221*
0
2
0
yydyy
5.0* Answer
From Blasius solution:
344.0*907.2
72.1
5
*
Boundary layer momentum thickness
0
1 dyUu
Uu
6321
0
2
32
0
2
2
0
yydy
yydy
yy
167.0 Answer
From Blasius solution:
133.0530.7
664.0
5
(b) From the definition of local skin friction:
U
y
u
y
0
0
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From the von Karman integral equation:
dx
dU
dx
dU
6
22
0
U
dx
dU
6
2
xU
dxU
dx
6
2
6 2
00
Uxx
U
x
x
x
U
x
12
1212
xxx Re
464.3
Re
12
Answer where:
Uxx Re
Drag per unit width: D/w
x
x
xxxx
xU
Ux
xUUxUx
U
U
dxx
U
Udx
U
x
Udx
UdxwD
Re
1
12
2
12
2
12
22
12
1212/
22
0
0000
0
2
1
2
1
xx
DxU
wDC
Re
1
464.3
4
Re
1
12
4/2
21
x
DCRe
155.1 Answer
Linear Approximation Blasius Solution 5.0* 344.0*
167.0 133.0
xx Re
464.3
xx Re
5
x
DCRe
155.1
x
DCRe
328.1
7/28/2019 ME2135 12/13 Tutorial Solutions 1-10
http://slidepdf.com/reader/full/me2135-1213-tutorial-solutions-1-10 23/23
For the velocity distribution:
2
yc
yba
U
u
2
2
cybU
dy
du
(a) First, find a, band cusing the boundary conditions: 0u at 0y (no slip condition) 0a
Uu at y (at the edge of boundary layer) cb1
0dy
duat y (at the edge of boundary layer) cb 20
Hence 2b and 1c and therefore the velocity distribution (profile) is given by:
2
2
22
yy
U
u
where:
ddyy
For 00 y
1 y
Displacement thickness:
1
0
32
1
0
2
0 3211*
ddyU
u
3
1* Answer
Momentum thickness:
5
11
3
51
5
1
3
5452
22422121
1
0
44321
0
432
1
0
433221
0
22
0
d
dddyU
u
U
u
15
2 Answer Hence, shape factor 5.2
*
H
Parabolic Approx. Blasius Solution
333.0* 344.0*
133.0 133.0
xx Re
47.5
xx Re
5
x
DCRe
46.1
x
DCRe
328.1