effect of different nose profiles on subsonic pressure coefficients ryan felkel department of...
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Effect of Different Nose Profiles on Subsonic Pressure Coefficients
Ryan Felkel
Department of Mechanical and Aerospace EngineeringCalifornia State University, Long Beach
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AIAA Student Research Conference Region VISan Diego, CA
March 24 – 26, 2011
Outline
Problem Description Pressure Gradients Munk Airship Theory Critical Pressure Coefficient Karman-Tsien Compressibility Correction Minimum Pressure Coefficient Critical Mach Number Summary
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Problem Description
Sounding rockets at high angle of attack shed a vortex pair from the forebody boundary layer• Especially when roll rate = pitch
natural frequency … called roll resonance, a high angle of attack flight condition
• Vortex pair induces a rolling moment
• Vortex-induced roll moment overrides both roll damping and driving (due to fin cant) torques
• Result is prolonged resonance, very high angles of attack, excessive drag and sometimes structural failure
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Problem Description, cont.
Boundary separation usually associated with an adverse pressure gradient ~ pressure increasing downstream Familiar examples
• Stalled wing• Behind a shock wave• Leeward side of a bluff body
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Low Pressure
High Pressure
Pressure Gradient Components
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• Circumferential Pressure Gradient (rB ∂p/∂Φ)• Flow around the body• Induced by angle of attack (α)• Unavoidable situation
• Longitudinal Pressure Gradient (∂p/∂z)• Flow along the axial frame of reference• Mitigation possible
0 20 40 60 80 100120140160180
-3-2-101Circumferential Pressure
Circumferential Angle (Φ)
Pres
sure
Co
effici
ent/
sin²
α0 10 20 30 40 50 60
-0.07
-0.05
-0.03
-0.01
0.00999999999999998
0.03
0.05
Longitudinal Pressure
Body Station, inches from Nose Tip
Prsu
ure
Coe
ffici
ent
Nose Profiles
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0 5 10 15 20 25 30 350
0.5
1
1.5
2
2.5
3
ConeOgiveOptimum
Body Station from Nose Tip [inches]
Body
Rad
ius
[inch
es]
Three nose shapes were analyzed: Cone, Ogive, and Optimum
Fineness ratio of 6 (L = 36”, D = 6”)
Munk Airship Theory
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r
z
R
zi
Process used to calculate pressure along an elongated airship with circular cross section
Points along the body station can be treated as three-dimensional source flows
Vector R is a position vector from a point along the body station axis to an off axis point
R moves along the surface of the nose to estimate the incompressible pressure coefficient acting on zi, which is the sum of all pressure coefficients with different R vectors
Different “zi”s are used to generate a Cp profile
VR
Munk Airship Theory is used to derive an equation for incompressible pressure coefficients based on source flow. (Derivation of equation found in paper.)
Munk Airship Theory Results
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0 10 20 30 40 50 60
-0.07
-0.05
-0.03
-0.01
0.00999999999999998
0.03
0.05
0.07
ConeOgiveOptimum
Body Station from Nose Tip [inches]
Pres
sure
Coe
ffici
ent [
Cp]
Fineness ratio of 6 (L = 36”, D = 6”)
Critical Pressure Coefficient
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0.600000000000001 0.800000000000001 1
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Free Stream Mach Number
Pres
sure
Coe
ffici
ent
Isentropic relation between P∞ and P* (M=1) Relation is plugged into the Cp equation with respect to free stream
Mach number Pressure Coefficient = (P*- P∞)/q
Karman-Tsien Compressibility Effects
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0 10 20 30 40 50 60 70 80 90 100
-0.035
-0.03
-0.025
-0.02
-0.015
-0.00999999999999999
-0.00499999999999998
1.38777878078145E-17
Incompressible Flow [M=0]Compressible Flow [M=0.6]
Body Station from Nose Tip [inches]
Pres
sure
Coe
ffici
ent [
Cp]
Optimum nose L= 36”, D= 6” cylindrical afterbody Incompressible pressure distribution from Munk Theory
0.850 0.870 0.890 0.910 0.930 0.950 0.970 0.990
-0.2
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
Critical Pressure CoefficientMinimum Compressible Cp
Free Stream Mach Number
Pres
sure
Coe
ffici
ent
Critical Mach Number Determination
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Critical Mach Number (M=0.95)
Critical Mach number for the nose cone occurs when critical pressure coefficient function and Karman-Tsien function intersect.
Minimum Incompressible Pressure Coefficient
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3.00 3.50 4.00 4.50 5.00 5.50 6.00
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
Cone-CylinderOgive-CylinderOpt-Cylinder
Nose Fineness Ratio [L/D]
Cp,m
in
Critical Mach Number
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3.00 3.50 4.00 4.50 5.00 5.50 6.000.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
Cone-CylinderOgive-CylinderOpt-Cylinder
Fineness Ratio [L/D]
Cri
tical
Mac
h N
umbe
r
Summary
The Experimental Sounding Rocket Association (ESRA) from CSULB will use this analysis for the 2011 Intercollegiate Rocket Engineering Competition (IREC) to design a nose for our bird, Gold Rush III
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Procedure was created to analyze different nose profiles with different fineness ratios
Mitigation of longitudinal pressure gradients Can be used for any continuous nose shape
Pressure Coefficient Pressure coefficient :
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2
21
V
PPCP
P = PressureV = Velocity
Use Bernoulli’s Law for pressure:2
1
V
VCP
Local Velocities can be found by solving LaPlace’s equation Since Laplace’s equation is linear, can superpose solutions Axial flow and cross flow solutions obtained separately and then
combined Evaluate velocity components on a body surface
Assume no boundary layer separation (e.g., α << 1) Axial flow solution
Free stream velocity = Tangency condition implies Vr = Use Munk Theory for Vz Circumferential symmetry implies Vθ = 0
cosV
dz
drV Bcos
Pressure Coefficient, Cont’d
Cross flow solution Free stream velocity = For a slender body, Vz = 0 Tangency condition implies Vr = 0 Circumferential velocity from doublet solution:
Full pressure coefficient on surface of a slender body (drB/dz << 1)
This leads to plots shown at the bottom of chart #3 16
sinV
sin*sin2 VV
22 sin41sincos2
V
VC zP