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]

Criti

cal M

ach

Num

ber

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 :

15

2

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VPPCP

P = PressureV = Velocity

Use Bernoulli’s Law for pressure:2

1

VVCP

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

dzdrV 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 #316

sinV

sin*sin2 VV

22 sin41sincos2

VVC z

P