FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional...
Transcript of FLOW NETS - Universiti Teknologi Malaysia · 2008-08-11 · FLOW NETS For any two-dimensional...
IDEAL FLOW THEORY
FLOW NETS For any two-dimensional irrotational flow of a ideal fluid, two series of lines may be drawn : (1) lines along which ψ is constant (2) lines along which φ is constant
SECTION B 1
IDEAL FLOW THEORY
stream line ψ perpendicular to the velocity potential φ These lines together form a grid of quadrilaterals having 90º corners. This grid is known as a flow net. It is provides a simple yet valuable indication of the flow pattern.
SECTION B 2
IDEAL FLOW THEORY
COMBINING FLOW
PATTERNS If two or more flow patterns are combined, the resultant flow pattern is described by a stream function that at any point is the algebraic sum of the stream functions of the constituent flow at that point. By this principle complicated motions may be regarded as combinations of simpler ones.
SECTION B 3
IDEAL FLOW THEORY
21 ψψψψ +∆+=AP
ψψψψ ∆++= 21AQ
The resultant flow pattern may therefore be constructed graphically simply by joining the points for which the total stream function has the same value. This method was first described by W.J.M.Rankine (1820-1872)
SECTION B 4
IDEAL FLOW THEORY
Velocity components ;
( ) 2121
21 uuyyyy
u +=∂∂
+∂∂
=+∂∂
=∂∂
=ψψψψψ
21 vvx
v +=∂∂
−=ψ
Net velocity potential ;
.......321 +++= φφφφnet
SECTION B 5
IDEAL FLOW THEORY
BASIC PATTERNS OF FLOW Uniform Flow ;
velocity components ;
αα
sincos⋅=⋅=
qvqu
stream function ;
vxuy −=ψ velocity potential ;
vyux +=φ
SECTION B 6
IDEAL FLOW THEORY
Source Flow ;
A source is a point from which fluid issues uniformly in all directions. If for two-dimensional flow, the flow pattern consists of streamlines uniformly spaced and directed radially outward from one point in the reference plane, the flow is said to emerge from a line source.
SECTION B 7
IDEAL FLOW THEORY
The strength m of a source is the total volume rate of flow from it. The velocity q at radius r is given by;
rmqπ2 velocitylar toperpendicu area
flow of rate volume==
velocity components ;
0
2
=∂∂
−=′
=∂⋅
∂=′
rv
rm
ru
ψθ
ψ
stream function;
πθψ
2m
source =
SECTION B 8
IDEAL FLOW THEORY
velocity potential ;
Crmsource += ln
2πφ ( I ) at 00,0 =⇒== Crφ
rmsource ln
2πφ =
( II ) at AmCAr ln2
,0π
φ −=⇒==
⎟⎠⎞
⎜⎝⎛=
Arm
source ln2π
φ
SECTION B 9
IDEAL FLOW THEORY
Sink ;
A sink, the exact opposite of a source, is a point to which the fluid converges uniformly and from which fluid is continuously removed. The strength of a sink is considered negative, and the velocities, ψ , φ are therefore the same as those for a source but with the signs reversed.
SECTION B 10
IDEAL FLOW THEORY
stream function;
πθψ
2sinkm
−= velocity potential ;
Crm+−= ln
2sink πφ ( I ) at 0,0 == rφ
rm ln2sink π
φ −= ( II ) at Ar == ,0φ
⎟⎠⎞
⎜⎝⎛−=
Arm ln
2sink πφ
SECTION B 11
IDEAL FLOW THEORY
Vortex ;
2 types ; 1. Irrotational vortex 2. Forced vortex
SECTION B 12
IDEAL FLOW THEORY
Irrotational vortex ; Circulation ;
δθδδ )(vortex rvvr ⋅′+′⋅=Γ
rv π2vortex ⋅′=Γ vorticity ;
0vortex =′
+′
=rv
rv
δδζ
stream function ;
rr
ln2vortex π
ψ Γ−=
velocity potential ;
θπ
φ2vortexΓ
=
SECTION B 13
IDEAL FLOW THEORY
Forced vortex ;
rv ⋅=′ ω vorticity ;
ωςζ 20 =⇒≠
rv
rv
δδωζ′
+′
==∴ 2
SECTION B 14
IDEAL FLOW THEORY
COMBINATION OF
BASIC FLOW PATTERNS Linear and Source ;
Stream function ;
sourcelinearncombinatio ψψψ +=
θπ
θψ ⋅+⋅−=2
sin mrUncombinatio
SECTION C 1
IDEAL FLOW THEORY
stagnation point S is the point where the resultant velocity is zero.
UmBOSπ2
== stream function at 0=θ ;
02
sin0 =⋅+⋅−== θπ
θψψmU
It is called ‘stagnation line’. The body whose contour is formed by the combination of uniform rectilinear flow and a source is known as a half body, since it has a nose but no tail, or Rankine body.
SECTION C 2
IDEAL FLOW THEORY
Distance from origin to 0=ψ ;
θπθsin2 ⋅
=Umr
Asymptote y ;
⎟⎠⎞
⎜⎝⎛−=⋅=
Um
Umry
2 and
2sinθ
Velocity components ;
θπ
cos2
⋅−=′ Ur
mu
θsin⋅=′ Uv
SECTION C 3
IDEAL FLOW THEORY
If rectilinear flow comes from the other side ;
2m
ncombinatio =ψ ( )
θπθπ
sin2 ⋅−
=U
mr
θπ
cos2
⋅+=′ Ur
mu θsin⋅−=′ Uv
SECTION C 4
IDEAL FLOW THEORY
Source and Sink ;
In this situation, the assumption again being made that the fluid extends to infinity in all directions.
SECTION C 5
IDEAL FLOW THEORY
Combination of stream function ; sinksource ψψψ +=ncombinatio
( )212θθ
πψ −=
mncombinatio
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
= −222
1 2tan2 yAx
Aymncombinatio π
ψ Component velocity ;
( ) ( ) ⎥⎦
⎤⎢⎣
⎡
+++
−+−
−= 22222 yAx
AxyAx
Axmuπ
( ) ( ) ⎥⎦
⎤⎢⎣
⎡
++−
+−= 22222 yAx
yyAx
ymvπ
velocity potential ;
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
2
1ln2 r
rmncombinatio π
φ
SECTION C 6
IDEAL FLOW THEORY
Source, Sink and Linear ;
Combination of stream function ;
linearncombinatio ψψψψ ++= sinksource
UyyAx
Aymncombinatio −⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−
= −222
1 2tan2π
ψ
SECTION C 7
IDEAL FLOW THEORY
Component velocity ;
( ) ( )U
yAxAx
yAxAxmu −⎥
⎦
⎤⎢⎣
⎡
+++
−+−
−= 22222π
( ) ( ) ⎥⎦
⎤⎢⎣
⎡
++−
+−−
= 22222 yAxy
yAxymv
π
value of x ;
1+=UAmAx
π value of ymax ;
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛= −
Ay
Umy max1
max tanπ
SECTION C 8
IDEAL FLOW THEORY
COMBINATION OF
BASIC FLOW PATTERNS Doublet ;
SECTION D 1
IDEAL FLOW THEORY
Stream function ;
( ) θπµθθ
πψ sin
22 21 rm
ncombinatio =−= velocity components ;
θπµ cos
2 2ru =′
θπµ sin
2 2rv =′
22 rq
πµ
= velocity potential ;
θπµφ cos
2 r−=
SECTION D 2
IDEAL FLOW THEORY
Doublet and Uniform ;
Stream function ;
θπµψ sin
2⎟⎠⎞
⎜⎝⎛ −= Ur
rncombinatio
SECTION D 3
IDEAL FLOW THEORY
velocity potential ;
θπµφ cos
2⎟⎠⎞
⎜⎝⎛ +−= Ur
rncombinatio
0=ncombinatioψ , 0=θ , πθ =
Ur
πµ
2=
SECTION D 4
IDEAL FLOW THEORY
UAr
πµ
222 ==
stream function ;
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2
2
1sinrAUrncombinatio θψ
velocity potential ;
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2
2
1cosrAUrncombinatio θφ
SECTION D 5
IDEAL FLOW THEORY
velocity components ;
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=′ 2
2
1cosrAUu θ
⎟⎟⎠
⎞⎜⎜⎝
⎛+=′ 2
2
1sinrAUv θ
velocity at cylinder surface ;
Ar = , °= 90θ 0=′u Uv 2=′
Pressure coefficient CP ;
θρ
22
21
12 sin41−=−
=U
PPCP
SECTION D 6
IDEAL FLOW THEORY
Pressure at cylinder surface ; ( )θρ 22
21
12 sin41−+= UPP Drag force FD ;
∫ =⋅= 0cosθdFFD Lift force FL ;
∫ =⋅−= 0sinθdFFL In real situation, both of these force are exist.
SECTION D 7
IDEAL FLOW THEORY
Doublet, vortex and Uniform ;
stream function ;
⎟⎠⎞
⎜⎝⎛Γ
−⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
Ar
rAUr C
ncombinatio ln2
1sin 2
2
πθψ
velocity components ;
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=′ 2
2
1cosrAUu θ
rrAUv C
πθ
21sin 2
2 Γ+⎟⎟
⎠
⎞⎜⎜⎝
⎛+=′
SECTION D 8
IDEAL FLOW THEORY
velocity at cylinder surface ;
Ar = , 0=′u
AUv C
πθ
2sin2 Γ
+=′ stagnation point S ;
Ar =
UAC
πθ
4sin Γ
−=
0sin0=
=Γθ
C
SECTION D 9
IDEAL FLOW THEORY
0sin0=
=Γθ
C
0.1sin4−<
<Γθ
πUAC
0.1sin4−=
=Γθ
πUAC
0.1sin4−>
>Γθ
πUAC
(Impossible)
SECTION D 10