School of Aerospace Engineering A Thesis Proposal by Ebru Usta Advisor: Dr.L.N.SANKAR APPLICATION OF...

School of Aerospace Engineering

A Thesis Proposal

by

Ebru Usta

Advisor: Dr.L.N.SANKAR

APPLICATION OF A SYMMETRIC TOTAL VARIATION DIMINISHING SCHEME

TO AERODYNAMICS AND AEROACOUSTICS OF ROTORS

Supported by the National Rotorcraft Technology Center(NRTC)


OverviewOverview• Motivation and Objectives • Background

• Mathematical and Numerical Formulation

•Symmetric TVD Scheme (STVD)

•Validation with 1-D and 2-D Wave Problem

• Results and Discussion

•Shock Noise Prediction for the UH-1H rotor

•Tip Vortex Structure and Hover Performance of the UH-60A rotor

• Proposed Work


MOTIVATION and OBJECTIVES

• Helicopter rotor’s flowfield is dominated by compressibility effects, a complex vortex wake structure and viscous effects.

• Accurate prediction of the aerodynamic flowfield and aeroacoustics of a helicopter rotor is a challenging problem in rotorcraft CFD.

• Existing methods for tip vortex and noise prediction suffer from numerous errors.

• As a result, accurate aerodynamics and aeroacoustics prediction methods are urgently needed.


PROBLEMS WITH THE CFD METHODSI. DISSIPATION ERRORS

• Numerical dissipation

–Dissipation causes a gradual decrease in the amplitude of an acoustic wave or the magnitude of the tip vortex as it propagates away from the blade surface.

–The computed vortical wake, in particular, diffuses very rapidly due to numerical dissipation


II. DISPERSION ERRORS

• Numerical dispersion

–Dispersion causes waves of different wavelengths originating at the blade surface to incorrectly propagate at different speeds.

–Because of dispersion errors, the waves may distort in nonphysical manner as they propagate away from the blade surface.


RECENT PROGRESS IN REDUCING DISPERSION ERRORS

• Tam and his coworkers recently developed a low dispersion numerical scheme called the Dispersion-Relation-Preserving (DRP) finite difference scheme(1996).

• Nance et. al. extended the DRP ideas to curvilinear grids(GT thesis 1997).

• Other works include: Carpenter, Baeder, Ekaterinaris, Smith et al. and CAA Workshops

I and II.


RECENT PROGRESS (continued)• Wang, Sankar and Tadghighi implemented

Nance's Low Dispersion Finite Volume (LDFV) ideas into TURNS and studied shock noise and hover performance of rotorcraft(1998).

–A side benefit of the high order accuracy LDFV and DRP schemes is their reduced dissipation or numerical viscosity.

–These schemes have numerical viscosity that is typically proportional to 5 where is the grid spacing.


RECENT PROGRESS IN REDUCING DISSIPATION ERRORS

• The easiest way to reduce dissipation errors is to increase the formal accuracy of the upwind scheme.

–Third order schemes in TURNS and OVERFLOW generate errors proportional to 3.

–Fourth order operator compact implicit schemes (OCI) have been studied by M.Smith (GT, 1994) and Ekaterinaris (Nielsen Eng.,1999)


RECENT PROGRESS (continued)

–Hariharan and Sankar have explored 5th order and 7th order upwind schemes with dissipation errors proportional to 5 and 7 respectively (GT thesis 1995).

–Wake studied the evaluation of a line vortex in space and time using 6th order spatially accurate scheme and have presented 9th order results in fixed wing mode(1995).


RECENT PROGRESS (cont’d)GRID CLUSTERING EFFECTS

• Numerical errors may also be reduced by use of a fine grid, and/or grid clustering.

–Tang et. al. recently have developed a grid redistribution method that clusters the grid points near the tip vortices and reduces the numerical diffusion of vorticity(1999).

–Strawn et. al. used high density embedded grids(CHIMERA) for improving the wake-capture (1999)


SCOPE OF THE PRESENT WORK

• The main purpose of this study is to develop and validate the spatially higher order accurate methods for modeling rotors in hover and forward flight.

• As the formal order of accuracy increases, it becomes more and more difficult to simultaneously reduce dispersion, dissipation and truncation errors.

• Are there better schemes available?


SCOPE OF THE PRESENT WORK

• Use Yee's symmetric TVD scheme to accurately model tip vortex structure and shock noise phenomena of rotors.

• Yee’s idea: High order central difference schemes can be coupled to lower order dissipation terms to yield accurate results.

• For this purpose, a version of the NASA Ames code TURNS, referred to here as TURNS-STVDx (x=4,6,8), has been developed.


WHAT IS A TVD SCHEME?

• For a TVD scheme, Sum of slopes always decreases, ensuring no new maxima occur.

nt

x

u

Sum of slopes = n

x

u

||

New Maximalnt

Sum of slopes =ln

x

u

||


Symmetric TVD Scheme

0

x

F

t

q

02/12/1

x

FF

dt

dq ii

The semi-discrete form at a typical node 'i' is:


Symmetric TVD Scheme (continued)

0)(||)(||

2

12/112/1

11

x

qqAqqA

x

FF

dt

dq

iiiiii

ii

• Dr. Helen Yee recommends the following second order form:

where computed using “Roe averages” of q at adjacent points.

||q

FA


STVD (cont’d)

Second order STVD scheme:

LRiii qqAqFqFF 2

1)()(

2

112/1

This part is used to control dispersion and truncation errors

This part is used tocontrol dissipationerrors

• Dispersion and dissipation errors may be independently controlled.


Fourth order STVD scheme:

LRiiiii qqAFFFFF 2

177

12

11122/1

STVD (cont’d)

and : MUSCL interpolation with a suitable limiter.Lq Rq

Sixth order STVD scheme:

LR

ii

iiiii

qqA

FF

FFFFF

2

1

8

37378

30

1

21

1232/1


STVD (cont’d)

Eighth order STVD scheme on Non-Uniform Grids:

LRiiii

iiiii

qqAhFgFfFeF

dFcFbFaFF

2

1)

(

4321

1232/1

distance along the coordinate line

21 ii xx2/1ix


STVD (cont’d)

– Where a,b,c,d,e,f,g,h are coefficients of the related fluxes.

• Note that this scheme also accounts for the non-uniform grid spacing.

4,..,2,3)(

)(

4

33

4

32/1

iiikxx

xx

a i

klil

li

i

klil

li

k


CONSTRUCTION OF and

• and were found using third order MUSCL interpolations.

• Koren Limiter, and a LDFV Limiter were explored.

• In some sample bench mark cases, and were found using higher order (4th, 6th and 8th) dissipation terms with no limiters.

LqRq

Rq Lq

Rq Lq


1-D WAVE PROBLEM

• The initial solution at t=0 is given by

• The exact solution is

0

x

u

t

u

16

2

)0,(x

etxu

16

2

),(tx

etxu


1-D WAVE PROBLEM (continued)

• The accuracy of the schemes is assessed by computing the of the error calculated as:

IMAX

uuErrorAverage

exact

2)(

IMAX : The maximum number of grid points

normL 2


1-D WAVE PROBLEM (cont’d)

• 1-D wave equation is solved explicitly using second order Runge Kutta method as follows:

n l l l nx

ni

pix u x u t u u) / (

plllpx

ni

pi

ni xuxutuuu )/(5.05.0)(5.01

l: Formal accuracy of the scheme


-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-3.2 -3 -2.8 -2.6 -2.4

LOG(1/N)

LO

G (

L2

NO

RM

)stvd4

stvd6

stvd8

• Higher order schemes, e.g. STVD8, consistently produces lowest errors on all grids.

• For STVD8, the slope is the steepest, indicating that the errors decrease quickly with refinement.


2-D Problem: Pulse interacting with uniform flow and solid wall.

CAA workshop test Problem organized by

Prof. Chris Tam (FSU)

0)()( '''

y

Bq

x

Aq

t

q

t=0+

V


• Several baseline solutions (6th order MacCormack, 3rd order Upwind) are available for comparison.

• Exact solutions are also available for comparison(Nance, Ph.D Dissertation)

• At boundaries, non-reflective boundary conditions were used.

• In this study,STVD4, STVD6 and STVD8 solutions were obtained. Only the 8th order results are shown here.

Approach:


BOUNDARY CONDITIONS

To avoid entropy layers, to preserve total enthalpy, h0

)( 'qAx

�

)()( '' qBqB yy

�

0)()( '''

y

Bq

x

qA

t

q

(No vorticity)

0)'(

0)'(

0

0

y

yp

yu

v


TIME HISTORY OF PRESSURE AT THE WALL

T=15

-0.01

0

0.01

0.02

0.03

0.04

0.05

-100 -50 0 50 100

mac4upwind3

stvd8exact

T=30

-0.1

0

0.1

0.2

0.3

-100 -50 0 50 100

mac4upwind3

stvd8exact

T=45

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-100 -50 0 50 100

mac4

upwind3

stvd8

exact

T=60

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-100 -50 0 50 100

mac4upwind3stvd8exact


T=75

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-100 -50 0 50 100X

P


T=100

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-100 -50 0 50 100


T=150

-0.1

-0.05

0

0.05

0.1

0.15

-100 -50 0 50 100

mac4

upwind3

stvd8

exact


PRESSURE CONTOURS

T=75

T=100

T=150

Oscillations due to no dissipation term

T=75


PRESSURE CONTOURS

Oscillations due to no dissipation term

With dissipation term

T=75T=75


PRESSURE CONTOURS(cont’d)

T=100T=100

With dissipation

OSCILLATIONS

T=100


PRESSURE CONTOURS(cont’d)

T=150 T=150

With dissipation


TRUNCATION ERROR ASSESMENT

T=15 T=30 T=45 T=60 T=75 T=100 T=150

UPW3 4.663E-05 8.028E-04 5.081E-04 4.523E-04 2.188E-04 1.822E-04 2.029E-04

MAC4 1.516E-04 3.479E-04 1.565E-04 1.282E-04 6.481E-05 2.139E-05 3.214E-05

STVD6 1.368E-04 2.704E-04 1.277E-04 1.110E-04 5.587E-05 2.715E-05 6.224E-05STVD8 1.461E-04 2.718E-04 1.232E-04 1.046E-04 4.541E-05 2.522E-05 4.939E-05

Scheme CPU TimeUPW3 16’:24MAC4 12’:12STVD6 11’:43STVD8 12’:15

CPU TIME:


RESULTS and DISCUSSION

• 4th,6th and 8th order Symmetric TVD schemes have been applied to model helicopter rotor shock noise for UH-1H rotor and tip vortex structure of UH-60A rotor.

• The following results are presented:

–Original TURNS code (3rd order MUSCL scheme)

–Modified flow solver TURNS-STVDx (x=4,6,8)

–Comparison with experimental data for UH-60A and UH-1H rotor.

• All rotor calculations were done on identical grids, to eliminate grid differences from skewing the interpretation of results.


SHOCK NOISE PREDICTION OF UH-1H ROTOR

• Calculations have been performed for a two-bladed UH-1H rotor in hover.

• The blades are untwisted and have a rectangular planform with NACA 0012 airfoil sections and an aspect ratio of 13.7133.

• The sound pressure levels have been compared to the experimental data for a 1/7 scale model (Purcell,1989).


Shock Noise Prediction, r/R=1.111, Tip Mach =0.90, Grid Size 75x45x31

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2000

0 0.5 1 1.5 2

Time (msec.)

Pre

ss

ure

(Pa

)

stvd6

exp

muscl

stvd4

stvd8

SCHEME (Pa-P)(Pascal)

% Error

Experiment -6302 0.00Baseline TURNS -5523 12.30TURNS-STVD4 -5536 12.15TURNS-STVD6 -5612 10.94TURNS-STVD8 -6311 0.14


Shock Noise Prediction, r/R=1.78, Tip Mach= 0.90,

Grid Size 75x45x31 -1600

-1400

-1200

-1000

-800

-600

-400

-200

0

200

400

00.511.52

Time(msec.)

Pre

ss

ure

(Pa

)

stvd6

exp

muscl

stvd4

stvd8


% Error



Shock Noise Prediction,r/R=3.09, Tip Mach =0.90,

Grid Size 75x45x31

-700

-600

-500

-400

-300

-200

-100

0

100

200

0 0.5 1 1.5 2

Time(msec.)

Pre

ssu

re(P

a)

stvd6

exp

stvd4

muscl

stvd8


% Error



PLANFORM OF THE UH-60A MODEL ROTOR

• Four blades, a non-linear twist, and no taper.

• 20 degrees of rearward sweep that begins at r/R=0.93.

• The aspect ratio and Solidity Factor 15.3 and 0.0825.


PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.920

r/R=0.920

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

Chord

-Cp

TURNS-STVD4

Experiment

TURNS-STVD6

TURNS-STVD8

TURNS


PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.99

r/R=0.99

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

Chord

-Cp

TURNS-STVD4

Experiment

TURNS-STVD8

TURNS-STVD6

TURNS


PERFORMANCE OF THE UH-60A ROTOR

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12Collective Pitch(deg.)

CT

/so

lidit

y

EXPERIMENT

TURNS-STVD4

TURNS-STVD6

TURNS-STVD8

TURNS


PERFORMANCE OF THE UH-60A ROTOR

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

CT/solidity

CQ

/so

lid

ity

EXPERIMENT

TURNS-STVD4

TURNS-STVD6

TURNS-STVD8

TURNS


PERFORMANCE OF THE UH-60A ROTORVISCOUS RESULTS for 149x89x61 GRID SIZE

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0 0.02 0.04 0.06 0.08 0.1 0.12CT/solidity

FM

experiment

TURNS-STVD4

TURNS-STVD6

TURNS-STVD8

TURNS


CONVERGENCE HISTORY FOR TURNS-STVD8 FOR UH-60A ROTOR

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0.0075

0 5000 10000 15000 20000

Iteration Number

CT


VISCOUS CALCULATIONS DONE IN COLLABORATION WITH UTRC AT UTRC ON A 181x75 x49 FINER GRID OF

UH-60A ROTOR

Blade Loading vs. collective pitch


Torque versus Blade Loading


Figure of Merit versus Blade Loading

Error of 0.01-0.02 in FM; well within 100 lb. or 200 lb. error in thrust; considered very good by industry.


CONCLUDING REMARKS

• The accuracy characteristics of the STVDx schemes have been systematically investigated in 1-D and 2-D problems where exact solutions exist.

• Several high order Symmetric TVD schemes have been implemented in the TURNS code .

• The tip vortex structure of UH-60A rotor and shock noise phenomena for UH-1H rotor are accurately modeled with these high order schemes compared to the baseline third order MUSCL scheme.


CONCLUDING REMARKS(cont’d)

• The eighth order STVD scheme is found to give the best thrust predictions for the UH-60A rotor, even on a coarse grid.

• The shock noise predictions were also, in general, better with the higher order schemes in spite of having loss in accuracy when a high scheme is used on a very coarse grid, 3 radii away.

• The STVDx schemes require little or no additional computational time, compared to the MUSCL scheme.


CONCLUDING REMARKS(cont’d)

• Many existing CFD solvers may easily be retrofitted with the symmetric TVD scheme.

• UTRC Viscous results compare very well with the model test.

• The Figure of Merit is generally 1-2 points under the experimental data which is considered very good.

• These results are much better than using baseline TURNS.


PROPOSED WORK

Perfecting the Hover Code:

• Increase formal accuracy of metrics, Jacobian, time, boundary conditions, load integration schemes.

• Additional validations for another rotor, to be chosen in consultation with industry and thesis committee.

• Study of Vortex Ring State and climb using GT experimental data


PROPOSED WORK (continued)

IF TIME PERMITSIF TIME PERMITS,

• Use embedded adaptive grid for improved wake capturing

• Use of Spalart-Allmaras turbulence model for hover prediction.

School of Aerospace Engineering A Thesis Proposal by Ebru Usta Advisor: Dr.L.N.SANKAR APPLICATION OF...

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