16.540 Spring 2006
PRESSURE FIELDS AND UPSTREAM INFLUENCE
1
PLAN OF THE LECTURE
•
Pressure fields and streamline curvature
– Streamwise and normal pressure gradients
– One-dimensional versus multi-dimensional flows
•
Upstream influence and component coupling
– How does the pressure field vary upstream of a fluid component
and when does this matter?
•
Pressure fields and the asymmetry of real fluid motions
2
NORMAL AND STREAMWISE PRESSURE GRADIENTS
•
•
Inviscid flow
Streamwise:
udu = − dp ρ
n
n
or ,
u
∂u 1 ∂p
=
∂l ρ ∂l
l
l
dl
l
dα
l+ dl
•
Normal
rc
u 2 1 ∂p
=
rc
ρ ∂n
3
dl
STREAMLINES AND WALL STATIC PRESSURES
B
Curved
wall
Pressure
A
Straight
wall
D
One-dimensional
C
A
D
n
n
C
B
4
ONE-DIMENSIONAL AND TWO-DIMENSIONAL DESCRIPTIONS
•
In a one-dimensional representation of a contraction, the pressure
gradient is always negative (or zero)
•
If adopt a higher fidelity (2D) description, this is not true
•
Is it possible to have a contraction in which there is no location that
has an adverse (non-favorable) pressure gradient?
•
Why is this important?
5
UPSTREAM INFLUENCE OF FLUID COMPONENTS
•
•
•
Approximate equation for the static pressure field
2-D, inviscid, steady flow, constant density
Velocity viewed as a uniform mean flow, uX , plus “small” nonuniformities, ux′ ,u y′ :
ux = ux + ux′
u y = u y′
•
Neglect products of small quantities in momentum equation
x - momentum :
u
1 ∂p ′
∂ux′
∂u ′
+ u y′ x = −
∂x
∂y
ρ ∂x
where p ′ is the departure from uniform static pressure
There is no term ux′
∂u
because u is uniform
∂x
6
So,
∂u x′
1 ∂p ′
=−
∂x
ρ ∂x
∂u ′y
1 ∂p ′
=−
ux
ρ ∂y
∂x
ux
and
Take
(a)
(b)
∂(a) ∂(b )
, yielding, using continuity
+
∂y
∂x
∂
0 =u
∂x
⎛ ∂u x′ ∂u ′y
+
⎜
∂y
∂
x
⎝
∇ 2p ′ = 0
⎞
1 ⎛ ∂ 2 p ′ ∂ 2 p ′⎞
⎟=− ⎜ 2 +
2 ⎟
∂
x
ρ
∂
y
⎠
⎠
⎝
Laplace' s Equation
• Famous equation with neat properties
• We will apply this to see the upstream influence
7
UPSTREAM INFLUENCE
•
Important question in internal flow systems– When are components coupled aerodynamcially
– When can they be considered independent?
•
•
•
Laplace’s equation gives direct and simple qualitative answer
Laplace’s equation gives direct and simple quantitative answer
First part:
- ∇ 2p′ has no intrinsic length scale
- If pick a y - length => x - length is set
- Consider an "unrolled" annular flow field
8
A DIGRESSION: WHAT DO I MEAN BY “LENGTH SCALE”?
•
What is an example of an equation with a length scale?
•
How about the momentum equation for viscous, constant-pressure
flow?
– This is an idealized example but it does make the point
•
Does this equation lead to some length scale, i.e. does a length
scale naturally arise out of the structure of the equation?
•
If so, what does it mean physically?
⎛ ∂ 2ux ∂ 2ux ⎞
∂ux
∂u x
ux
+ uy
= ν⎜ 2 +
⎟
∂x
∂y
∂y 2 ⎠
⎝ ∂x
9
BACK TO LAPLACE: RELATION OF
CIRCUMFERENTIAL AND AXIAL LENGTH SCALES
•
Consider an “unrolled” annular flow field, mean radius rm
•
Suppose we have a circumferential length scale rm/n, then the axial length
scale (extent of upstream influence) is also rm/n
•
Circumferential length scale sets the region of upstream influence
•
Will show this explicitly in two examples:
•
Upstream influence of a circumferentially periodic non-uniformity
•
Upstream influence of a radially non-uniform flow
10
y (Circumferential Direction)
Uniform flow
far upstream
(x -∞)
Static pressure
nonuniformity
defined at x = 0
U
Background
(mean) velocity
Components due
to nonuniformity
W (Blade Spacing)
x
Flow domain used in estimation of upstream influence region for periodic array
(turbine blading); region of interest is x < 0.
Figure by MIT OCW.
11
UPSTREAM INFLUENCE OF A CIRCUMFERENTIALLY NONUNIFORM FLOW
•
•
Blade row with blade-to-blade spacing W
Whatever the loading distribution (compressor, turbine, pump) the
static pressure distribution along x=0 can be written as
- p ′(0,y ) =
∞
∑ (a e
2 πiny / W
n
)
; Fourier series
n=-∞
- To match this boundary condition, p ′(x,y ) must also
be of this y - dependence
- Also p ′ (static pressure non- uniformity) must be bounded
far upstream
- Thus
p ′(x,y ) =
∞
∑ f (x )[a e
n
2 πiny / W
n
]
n=-∞
-
Plugging in to Laplace, fn (x ) is found to have the form of
exponentials : e 2 πnx/W and e −2 πnx/W
12
•
•
Physical solutions must be bounded far upstream
Thus, only positive exponentials allowed
p ′(x,y ) =
∞
∑e
2π n x / W
[a e
n
2 πiny / W
]
n=-∞
•
Lowest harmonic (often, but not always, n=1) has the largest
upstream influence
2π n x / W
p ′(x,y ) ∝ e
[a1e 2 πiny / W + a−1e −2πiny / W ]
p ′(x,y ) = p ′(0,y )e 2 π n x / W
•
Exponential decay
Different phenomena have different upstream extents of influence
13
FEATURES OF THE SOLUTION
•
•
We neglected nonlinear terms: are they important over the domain
of interest or, more precisely, for the problem of interest?
What else did we neglect?
•
At a distance W/2 upstream the non-uniformity is 0.04 its value at
x=0
•
Blade spacing length scale is W
•
Inlet distortion length scale is radius of the compressor
– Upstream effects are much stronger
•
What can we state about applicability and limits of the conclusions
from this simple analysis?
14
INSTRUMENTATION FOR TF30 ENGINE TEST
Front Bulkhead
Rotating Screen
High- Pressure
Compressor
Lab Seal
2A 2B
A 2
Station 1
(2 Rows of
Static Taps)
Fan
Low-pressure
compressor
IGV (STA. 2C)
Exhaust Nozzle
o
o
315
270
90
o
255
o
o
108
180
o
o
345
o
o
185 180o
135
o
140
o
280
o
o
80
o
90
o
100
o
270
o
260
225
Station 2A - 44.41 cm (17.48 in)
(Fan Inlet) Upstream of IGV Midspan
0 IGV
o
(23.5 )
1
2 3
o
(39.2 )
4
5
o
o
47
o
62.5
90
o
o
90
o
95
o
45
o
190
o
o
180 170
o
135
o
Station 1.0 (Airflow Metering Station)
250.24 cm (98.52 in.) Upstream of IGV
Midspan
o
0
o
270
o
242.5
o
225 o
207.5
o
o
Station A (Behind Rotating
Screen Assembly) 83.55 cm
(32.90 in.) Upstream of IGV
Midspan
332
o
315
o
288
275
o
270
o
350 0 10o
180 165
117.5
o
120
o
152.5
o
Station 2 (Fan Inlet) 13.39
cm (5.27 in.) Upstream of
IGV Midspan
o
90
o
180
o
Station 2C (IGV Inlet)
*Steady-State Total Temperature
o
Hi-Response Total Temperature
Steady-State Static Pressure
Steady-State Total Pressure
Pitch-Yaw Pressure Probes
Boundary-Layer Yaw Probes
Figure by MIT OCW.
15
BULLET NOSE EXTENSION WITH PRESSURE TAPS
Sta. 2B static taps located at
22o and 6.17 cm (2.43 in.)
upstream of the IGV's
Steady-state
static pressure
Rotating screen
Extra length
assembly
bullet nose
Fan
Station 2
instrument plane
Two axial rows of static
pressure taps. Each row contains
15 pressure taps located at 120o
(looking upstream)
IGV (STA. 2C)
Figure by MIT OCW.
16
VARIATION OF STATIC PRESSURE WITH
DISTANCE UPSTREAM OF AXIAL COMPRESSOR
[Soeder and Bobula]
Static pressure distortion level,
|∆p/∆pinlet|
Screen Blockage
26.0%
34.6%
49.4%
exp[-(x-x2B)/R]
0.8
0.4
Sta. 2B
0
0
Screen
1.0
2.0
3.0
Normalized distance from IGV, x/R
Figure by MIT OCW.
17
4
Compressor
Compressor
Blade-row model
L
H
Inter - Spool
gap
L
1
-∞
3
OGV
Mean
radius
r
2
IGV
OGV
1
CL
Actuator disk
model
2
x
=0
r
H
3
4
x
=L
r
+∞
Two spool arrangement
Compressor Coupling in Two Shaft Engine
Figure by MIT OCW.
[Williams and Ham]
18
Rings of Static Tappings
CMS
B
2
C
2
Tip radius
15.24 CMS
D
2
E
2
F
2
G
H
2
Static Tappings J
2.75
I
2.75
0.75
7.5
Eight
Airflow
Vanes
Hub radius
10.41 cms
16
17 18
1
θ
2
15
3
14
4
13
5
12
11
CL
Distortion
generator
Distortion
generating blockage
(Full annulus
blockage of
90o extent)
6
7
10
9 8
Circumferential location of
instrumentation, vanes &
blockage
Duct centerline
Scale
5cm
Model Test Geometry and Instrumentation
Figure by MIT OCW.
[Williams and Ham]
19
UPSTREAM DECAY OF STATIC-PRESSURE DISTORTION
Experiment
Measured CP
Ring I
0.4
0.2
Fourier Fit
1.0
CP 0
0.8
Prediction using
Ring I as input
-0.2
0.6
-0.4
(CP)LOCAL
(CP)I
Ring D
0.2
Prediction
CP 0
-0.2
0.4
0.2
2
6
10 14 18
Angular position
0
B
C
D
E
F
G
H
I
Axial ring location
Figure by MIT OCW.
[Williams and Ham]
20
FEATURES OF THE UPSTREAM FLOW FIELD
(Low Mach Numbers)
• Inlet distortion (non-uniformity with length scale R)
• Total pressure constant along streamlines (Bernoulli)
2
• Static pressure obeys Laplace’s equation: ∇ p ′ = 0
1 ∂ 2p ′
rm
2
∂ 2p ′
+
=0
2
2
∂θ
∂x
;
p ′ is static pressure disturbance
• No inherent length scale in equation
• Length scale set by boundary conditions
• If θ - length scale is ~ L then upstream decay of static pressure
−2π x / L
is like e
• Region of influence ~ diameter
21
Tunnel Speed = 147 knots
Engine mass flow = 36.1 lb/s/ft2
1
3
5
1
7
9
+30o
Incidence =
without engine interaction
Local total pressure
recovery
6 = 0.90
1 = 0.99
7 = 0.86
2 = 0.98
8 = 0.82
3 = 0.96
9 = 0.78
4 = 0.94
0 = 0.74
5 = 0.92
7
Incidence = +35o
with engine interaction
Short Pitot Intake Effect of Presence of an Engine
Figure by MIT OCW.
[Hodder]
22
UPSTREAM INFLUENCE OF A RADIALLY NON-UNIFORM
ANNULAR FLOW
•
Again have uniform background flow in x-direction, ux′ ,ur′ << ux
•
Pressure varies with radius at x=0 (boundary condition)
∂
= 0 ; axisymmetric variation
∂θ
∂( )
1 ∂p ′
∂ux′
ux
=−
; take
∂x
ρ ∂x
∂x
1( ) ∂ ( )
1 ∂p ′
∂ur′
ux
=−
; take
+
∂x
ρ ∂r
r
∂r
∂ 2 p ′ 1 ∂p ′ ∂ 2 p ′
+
+
=0
2
2
∂r
r ∂r ∂x
;
∇ 2p ′ = 0
23
ro
(outer radius)
r
u'r
Background
(mean) velocity
U
u'x
Components due
to nonuniformity
ri
(inner radius)
x
∆r
Static pressure
nonuniformity defined
at x = 0
CL
Annular flow geometry used in estimation of upstream influence region for
axisymmetric flow; region of interest is x < 0.
Figure by MIT OCW.
24
•
•
•
Suppose annulus has a high hub/tip radius ratio
Length scale of non-uniformities is
∆ r = ro − ri
Ratio of
∂ 2p ′
∆r
1 ∂p ′
to
is
<< 1
rm
∂r 2
r ∂r
∂ 2p ′ ∂ 2p ′
Reduces to
+
≅0
∂r 2 ∂x 2
Solution is p ′ ∝ exp (−π x / ∆r )
Can extend to compresible flow using Prandtl- Glauert
transformation : x → x 1− M 2
25
INTERACTION LENGTH, L, FOR FLOW
NONUNIFORMITIES
2π
Unwrapped
compressor
L
L
Tip
L
0
Circumferential distortion
~ Diameter
Hub
Radial Distortion
~ Span
Figure by MIT OCW.
Blade-to-Blade
Nonuniformities
~ Gap (Chord)
26
INTERACTION BETWEEN COMPONENTS
SCREEN AND CONTRACTION
x
Flow
W1
Uniform
screen
W2
y
Downstream (2)
Upstream (1)
Figure by MIT OCW.
27
PRESSURE FIELD AT DIFFERENT AXIAL STATIONS
Top of duct
p(x,y) - p
1 ρu 2
x2
2
-1
0
1 -1
0
1 -1
0
1 -1
0
1
Bottom of duct
Flow
Figure by MIT OCW.
28
WHAT DO WE EXPECT THE INTERACTION TO DO?
• What effect does a screen have on a non-uniform flow?
• How would you characterize the attributes of a screen?
29
WHAT DO WE EXPECT THE INTERACTION TO DO?
• What effect does a screen have on a non-uniform flow?
• How would you characterize the attributes of a screen?
• In regions in which the velocity is high what is the local
pressure drop through the screen?
• Where are the regions (across the channel) in which the
velocity is high?
30
IMPACT ON DOWNSTREAM CONDITIONS
1.25
Downstream velocity
Screen-contraction spacing = 0
0.5
0.25
0.125
ux
ux2
1.0
0.75
0.5
pt - ptcenterline
1 ρux2
2
2
Downstream stagnation pressure
Screen-contraction spacing = 0
0.25
0.125
0.0
-0.4
-0.2
0.5
0.25
0.0
0.2
0.4
y / W2
Figure by MIT OCW.
31
WHAT IS THE EFFECT OF A SCREEN ON A NONUNIFORM UPSTREAM FLOW?
y
Screen
Region I
W
Region II
0
x
Streamline
Figure by MIT OCW.
32
DOES THIS EFFECT DEPEND ON THE SCREEN
CHARACTERISTICS? HOW?
• Does the pressure field upstream of the screen depend
on the pressure drop through the screen?
• What are the features of the upstream pressure field
• For a given far upstream flow non-uniformity, would we
get a more uniform velocity downstream if the pressure
drop increased?
• How do we connect the pressure field to the velocity
non-uniformity?
- Upstream?
- Downstream?
- Upstream to downstream?
33
EFFECT OF SCREEN PRESSURE DROP ON
DOWNSTREAM VELOCITY (I)
1.0
Data reported by Taylor and Batchelor (1949)
Data of Davis (1957)
0.5
Calculation based on η = 0
(axial flow downstream of screen)
u'xd ( ,y)
u'xu(- ,y)
0.0
-0.5
Screen pressure
10 drop coefficient, K
0
Calculation based on η given by
Eq. (12.2.18) (refraction by screen)
Figure by MIT OCW.
34
EFFECT OF SCREEN PRESSURE DROP ON
DOWNSTREAM VELOCITY (II)
Low Pressure Drop Screen
High Pressure Drop Screen
1.5
Velocity / mean velocity
1.25
1.0
0.75
0.5
0.25
Upstream velocity
Measurements
Calculation
0
-0.5
Downstream
velocity
0
Distance / channel width
0.5
-0.5
0
0.5
Distance / channel width
Figure by MIT OCW.
Low pressure drop screen
High pressure drop screen
35
REGION OF INFLUENCE OF SCREEN
[K is screen pressure drop]
1.0
0.5
K=1
uxk(x,y)
uxk(
,y)
K=2
K=5
K=3
K = 10
0.0
-1.5
0.0
kx/W
1.5
-0.5
Figure by MIT OCW.
36
Pressure coefficient, Cp
A MORE COMPLICATED (OR IS IT?) EXAMPLE: STATIC PRESSURE
FIELD UPSTREAM OF A COMPRESSOR STATOR/STRUT GEOMETRY
[The figure shows only one section (one “wavelength”) of a periodic geometry]
0.2
Analysis
Measurements
0.0
-0.2
10
30
50
0
25
50
75
Circumferential distance (% strut gap)
100
Figure by MIT OCW.
37
WHAT ARE THE FEATURES OF THIS PRESSURE FIELD?
(Location is 0.5 of a stator chord upstream of the stator leading edge)
• The figure shows a section of a periodic geometry. The
geometry is repeated (on both sides) to simulate a full
annulus with, say, twelve struts and 72 blades
• There is a large length-scale non-uniformity
• There are smaller length scale “bumps” on this
• How would we explain the features of this pressure field?
38
Inflow & Outflow to Fluid Devices Asymmetry of Real Fluid Motion
•
Examples we looked at had fundamental asymmetry
(inflow to inlet: streamlines entered from all directions - however
outflow of ejector: parallel jet exited in direction of exit nozzle)
•
Different streamlines configuration associated with inflow or exit
flow!
Note: asymmetry was implicit in control volume treatment
•
Cause for asymmetry
no-slip condition at solid surface,
feature of all real fluids!
•
For high Re-flow:
u
∂u
1 ∂p
=−
∂x
ρ ∂x
du = −
dp
ρu
so for same ∆p, higher u yields smaller ∆u
39
Boundary Layer Subjected to Pressure Rise
•
Same ∆p in free-stream as in BL
•
Fluid in BL retarded by viscous forces (a) to (b) to (c)
•
For same pressure rise, BL suffers larger drop in velocity than fluid
outside BL
flow will eventually separate
Separation
streamline
(a)
•
(b)
(c)
The ∆p at which ∆u is driven to 0 in BL is less than that which freestream can attain
40
Contrast Between Inflow to, Outflow from, Pipe
•
Inflow:
– favorable pressure gradient from 1 to
2 (acceleration of fluid in BL)
– at 2 have region of low pressure
(streamline curvature)
②
– from 2 to 3 static pressure rises again,
BUT outside boundary layer (BL)
some streamline convergence, so
severity of adverse pressure gradient
③ ①
lessened
– adverse pressure gradient mild
enough to avoid separation
For high Re, BL thin
streamlines for flow into pipe will follow geometry
41
Contrast Between Inflow & Outflow of Pipe
•
How about outflow of pipe:
– will outflow have also this streamline configuration?
– why or why not? What are pressure gradients driving the
outflow pattern?
•
Asymmetry in streamline configuration due to viscosity
•
Motions are not reversible in thermodynamic and kinematic sense
External Flow Example:
thin wing: Kutta-Joukowski
condition
Internal Flow Example:
flow leaving straight nozzle,
parallel to nozzle axis
Good assumptions for describing real flow
42
Example: Flow Through a Bent Tube
•
Freely rotating bent tube,
constant area A, volume rate of
flow Q
•
Flow entering at center 0 and
exiting through bent part
•
What is rotation rate Ω ?
•
What happens if there is inflow instead of outflow through bent
Figure by MIT OCW.
tube?
•
Does it rotate? Why or why not?
43
FLOW INTO (a) AND OUT OF (b) A PIPE IN A QUIESCENT FLUID
u=0
p = p∞
Outflow
velocity
Inflow
velocity
Pipe
Pipe
(a) Inflow
(b) Outflow
Figure by MIT OCW.
44
INFLOW FROM A QUIESCENT FLUID INTO A PIPE:
FLOW NEAR THE PIPE ENTRANCE
②
③ ①
45
EXIT FLOW FROM A SUBSONIC NOZZLE WITH PRESSURE
INSIDE THE JET HIGHER THAN pambient
phigh
Uniform p?
pambient
Figure by MIT OCW.
46