School of Aerospace Engineering APPLICATION OF A SYMMETRIC TOTAL VARIATION DIMINISHING SCHEME TO...

School of Aerospace Engineering

APPLICATION OF A SYMMETRIC TOTAL VARIATION DIMINISHING SCHEME TO

AERODYNAMICS OF ROTORS

Supported by the

United Technologies Research Center Internship

National Rotorcraft Technology Center (NRTC)

Nielsen Engineering and Research

Ebru Usta

Advisor: Prof. Lakshmi N.Sankar


OVERVIEWOVERVIEW• Background • Previous Work

• Research Objectives

• Mathematical and Numerical Formulation

• Results & Discussion

1-D Model Problem

Two-Bladed Caradonna-Tung Rotor

UH-60A Model Rotor

• Conclusions and Recommendations


PROBLEM IN HOVERPROBLEM IN HOVER

• Design for Hover on a hot day, out-of-ground effect is an important area

THRUST

POWER

Light Loads

Heavy LoadsArea of importance to Rotorcraft Designers

A

ThrustThrustPower

2 A:Rotor Disk Area


Momentum Theory

• Developed by Rankine and Froude

• Gives back-of-the-envelope estimates for power:

where T= Thrust, A= Disk Area, P= Power

• Can not address details of the rotor, number of blades, etc.

A

TTP

2


Blade Element-Momentum Theory

• Based on Prandtl’s lifting line theory, extended to propellers by Goldstein.

• Combines inflow velocities from annular momentum theory, and 2-D airfoil aerodynamics.

• Can handle details of the blade geometry, but requires a table-look up of airfoil load characteristics.

• Requires empirical models for tip loss, profile drag loss, swirl.


Vortex Theory

• Proposed by Gray in 1950s, perfected by Landgrebe and others during the 70s and 80s.

• Uses an assumed or computed wake geometry, and Biot-Savart Law for inflow.

• Retains many of the features and limitations of BEM theory.


CFD Based Methods for Hover• Potential Flow based Methods were developed during the mid

1980s. (Sankar and Prichard; Strawn; Steinhoff and Ramachandran).

• Euler and Navier-Stokes based methods with a prescribed inflow during late 1980s. ( Wake and Sankar, Agarwal,..)

• Vortex capturing Navier-Stokes methods became available during the 1990s (Srinivasan and Baeder, Hariharan).

• Work also began on improving the solution accuracy (Smith,Hariharan, Baeder and Tang) .


MOTIVATION BEHIND THE WORK

• The predictive capability of modern CFD methods is still unsatisfactory.

• Many industry standard codes (e.g. TURNS, OVERFLOW) have low order spatial accuracy which leads to excessive numerical diffusion and dispersion in the solution of the flow field.

• A very fine grid, and large CPU resources are needed to reduce these errors.

• High order algorithms are an effective way of reducing these errors and achieving accurate solutions on moderately fine grids.


RECENT PROGRESS IN REDUCING DISSIPATION ERRORS

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

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

– M.Smith (1994) and Ekaterinaris (1999): 4th order operator compact implicit schemes (OCI)

– Hariharan and Sankar (1995): 5th order and 7th order upwind schemes.

– Wake et. al. (1995): 5th-9th order spatial accuracy schemes in fixed and rotary-wing modes.

• Dispersion errors are also simultaneously reduced.


RESEARCH OBJECTIVES

• Develop and evaluate high order spatial accuracy schemes for modeling rotors in hover.

• Develop compatible numerical dissipation models, that permit accurate vortex capturing.

• Investigate compatible approaches for metrics calculations, load integration, and turbulence modeling.

• Validate the algorithms and approaches and draw conclusions on what works, and why.


Model Problem

1-D Vorticity Transport Equation:

0

xc

t

: Vorticity

c: Convection Speed

01

111

xc

t

ni

ni

ni

ni

Propagation Direction

i -1 i


• Low order upwind scheme is very dissipative. There is a reduction in the amplitude of the wave.


Let’s revisit 1-D Vorticity Transport Equation:

0

xc

t

212

2 2x

x

x x xi i i i i i i

Symmetric Part Numerical Viscosity

• If we replace these low order schemes with high order counter parts, results dramatically improve.


•High order scheme is less dissipative and less dispersive, compared to the low order upwind scheme.


High Order Accuracy Schemes for System of Equations

x

F

x

FF 2/1i2/1i

2/1iF Symmetric Part + Numerical viscosity

Second order

Fourth order

Sixth order

Eighth order

First order

Third order(MUSCL)

Fifth order (WENO)

(Base Scheme) (Filter)

• Symmetric Schemes have no built-in numerical viscosity. • Needed to be added explicitly.


STVD FRAME WORK

2/1iF Symmetric Part

i1i FF2

1

1ii1i2i FF7F7F12

1

2i1ii1i2i3i FF8F37F37F8F30

1

LR qqA2

1

2nd order:

4th order:

6th order:

This part is used tocontrol dissipationerrors

8th order:

321

1234

329139533

533139293

840

1

iiii

iiii

FFFF

FFFF


STVD FRAME WORK

2/1iF Symmetric Part LR qqA2

1

3rd order MUSCL:

LEFT STENCIL

RIGHT STENCIL

RqLq

i-1 i

i+1/2

i+1 i+2

Cell Face

1st order MUSCL:


STVD FRAME WORK

2/1iF Symmetric Part LR qqA2

1

5th order WENO:i

RqLqLEFT STENCIL

i+1/2 Cell Face

i-1 i+1 i+2 i+3i-2

)3(2/13

)2(2/12

)1(2/11

~~~ iiiR qrqrqrq

,ˆ )3,2,1(2/1iq

)3,2,1(2/1

~iq : Three estimates for

: Nonlinear Weights ,,, 321 www 321 ,, rrr

LEFT STENCIL

Lq


How well do these schemes work?

• To answer these questions:

– 1-D model problem is revisited.

– Next, these schemes are applied to rotors in hover.


Error Analysis of the1-D Model Problem considered earlier

• Slope of high order schemes is high, meaning errors are rapidly reduced as the grid is refined.

• High order schemes have consistently lower errors, except on very coarse grids.

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-04 1.E-03 1.E-021/N

L2-Norm

2nd order Symmetric Scheme

4th order Symmetric Scheme




Dispersion Characteristics of High Order Schemes

Vertical Axis: Ratio between a numerical approximation to u/ x and the exact expression

for the trial function u=exp(ipx)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Wave Number*Grid Spacing

Ratio

2nd order Symmetric4th order Symmetric6th order Symmetric8th order SymmetricFirst order UpwindThird order Upwind 3 points


Hover Simulations: Present Contributions

• Research performed included:– Implementation of high order symmetric

schemes in an industry standard methodology– Implementation of high resolution WENO

scheme– Implementation of high accuracy metric

calculations, and load integration– Evaluation of advanced turbulence model– Code validation and interpretation of physical

features.


TURNS• TURNS: A public domain Transonic Unsteady

Rotorcraft Navier-Stokes code.• It solves the strong conservation form of 3-D Navier-

Stokes equations using a Finite Volume scheme. • TURNS uses LU-SGS implicit time marching scheme.• It has an option to use Newton sub-iteration at each

time step to reduce the linearization and factorization errors and improve the temporal accuracy.

• The inviscid fluxes crossing the cell faces are computed by ROE scheme.

• 3rd MUSCL interpolation is used to interpolate the flow properties stored at cell centers to cell faces.

• Different type of limiter options are available across high gradients.


CH-GRID TOPOLOGY(149x89x61=800K)

2R

2R

BLADE

4-bladed rotor

2-bladed rotor


TIP AND WAKE MODIFICATION IN THE GRID

Clustering does not follow the tip Clustering at the tip

ORIGINAL GRID GENERATION MODIFICATION

BLADE TIPBLADE TIP

Oscillations Smooth lines


RESULTS- I• Caradonna-Tung Rotor

–Two bladed, untwisted, rectangular blade

– AR = 6, NACA 0012 airfoil sections

–Tip Mach number= 0.44, 8 degree collective pitch

–Data available includes pressures, total thrust, and wake geometry.

• A number of combinations of the algorithm were studied.

– STVDx: x=4,6,8

– MUSCL vs. WENO 5


Surface Pressure Distributionsr/R=0.89

-1.3

-0.8

-0.3

0.2

0.7

1.2

0 0.5 1

x/cCp

EXPBaseline (MUSCL3+BL)STVD4+MUSCL3+BLSTVD6+MUSCL3+BLSTVD8+MUSCL3+BL

r/R=0.96-1.35

-0.85

-0.35

0.15

0.65

1.15

0 0.5 1

x/c

Cp

EXPBASELINE (MUSCL3+BL)STVD4+WENO5+BLSTVD6+WENO5+BLSTVD8+WENO5+BL

•All the methods predict reasonable Cp distributions


Surface Pressure Distributions

• All the methods predict reasonable Cp distributions

• However, differences between the various schemes become more apparent when they are integrated

r/R=0.89-1.3

-0.8

-0.3

0.2

0.7

1.2

0 0.5 1

x/cCp

EXPBaseline (MUSCL3+BL)STVD4+MUSCL3+BLSTVD6+MUSCL3+BLSTVD8+MUSCL3+BL

r/R=0.96-1.35

-0.85

-0.35

0.15

0.65

1.15

0 0.5 1

x/c

Cp

EXPBASELINE (MUSCL3+BL)STVD4+WENO5+BLSTVD6+WENO5+BLSTVD8+WENO5+BL


Radial Loads

• STVD schemes are seen to improve the results compared to baseline scheme, but benefits taper off • When large stencil schemes are used on highly stretched grids

0.1

0.15

0.2

0.25

0.3

0.35

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Radial Station,r/R

Sectional Thrust Coefficient

EXP

BASELINE (MUSCL3+BL)

STVD4+WENO5+BL

STVD6+WENO5+BL

STVD8+WENO5+BL


Wake Geometry: Contraction Rate

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 45 90 135 180

Vortex Age, degrees

r/R

ExperimentBASELINE (MUSCL3+BL)STVD8+WENO5+BLHariharan [60]


Wake Descent Rate

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 50 100 150

Vortex Age, degrees

z/R

Experiment

Baseline (MUSCL3+BL)

STVD8+WENO5+BLHariharan Ref [60]


ISO VORTICITY CONTOURS

• These qualitative pictures are shown for three reasons:

1. To show the tip vortex shedding from both different algorithms,

2. To show the ability of the WENO scheme over MUSCL scheme since It does not diffuse the tip vortex as much as MUSCL scheme.

3. To show the ability of the WENO scheme to capture the first,second, and third passages of the tip vortex.

Tip Vortex

Vortex Shedding

First Passage

Second Passage


VORTICITYCONTOURS IN THE r-z PLANE


RESULTS-II• A number of combinations of the algorithm were studied

and applied to UH-60A rotor.

–STVDx: x=4,6,8

–3rd order MUSCL vs. 5th order WENO5

–Second order vs. Fourth order Metrics

–Second order vs. Fourth order Load integrations

–Baldwin-Lomax Turbulence Model vs. Spalart-Allmaras

• Pressure distributions and performance predictions are presented for this rotor and compared with experimental data.


PLANFORM OF THE UH-60A MODEL ROTOR

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

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

• The aspect ratio and solidity: 15.3 and 0.0825.


PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.920, Grid Size 149x89x61(808K)

r/R=0.920

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

x/c

Cp

ExperimentBASELINE (MUSCL3 + BL)STVD4 + MUSCL3 + BLSTVD6 + MUSCL3 + BLSTVD8 + MUSCL3 + BL

r/R=0.920

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

x/c

Cp

ExperimentBASELINE (MUSCL3 + BL)STVD6 + WENO5 + BLSTVD8 + WENO5 + BL

• All methods give reasonable Cp distributions


PRESSURE DISTRIBUTION ALONG THE SURFACE OF UH-60A AT r/R=0.99, Grid Size 149x89x61(808K)

r/R=0.990

-1.7

-1.2

-0.7

-0.2

0.3

0.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/c

Cp

ExperimentBASELINE (MUSCL3 + BL)STVD4 + MUSCL3 + BLSTVD6 + MUSCL3 + BLSTVD8 + MUSCL3 + BL

r/R=0.990

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0

x/c

Cp

ExperimentBASELINE (MUSCL3 + BL)STVD6 + WENO5 + BLSTVD8 + WENO5 + BL

• Overprediction of the suction peaks near the tip is related to the inaccurate strength and location of the passing vortex.


EFFECTS OF THE SYMMETRIC PART OF THE ALGORITHM ON THE PERFORMANCE OF THE UH-60A

ROTOR GRID SIZE 149x89x61(808K)

0.005

0.007

0.009

0.011

0.065 0.09 0.115

CT/

CQ/

ExperimentBASELINE (MUSCL3+BL)STVD4+MUSCL3+BLSTVD6+MUSCL3+BLSTVD8+MUSCL3+BL

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.025 0.05 0.075 0.1 0.125

CT/

CQ/


• Prediction steadily improves as the symmetric part of the algorithm changes from 4th, 6th to 8th.•The slope of the baseline is incorrect. It underestimates the required torque at higher thrust settings.


PERFORMANCE OF THE UH-60A ROTOR GRID SIZE 149x89x61(808K)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.025 0.05 0.075 0.1

CT/

FMExperimentBASELINE (MUSCL3+BL)STVD4+MUSCL3+BLSTVD6+MUSCL3+BLSTVD8+MUSCL3+BL

0.55

0.6

0.65

0.7

0.75

0.05 0.075 0.1

CT/

FM


• As blade loading increases, the STVD8 predictions is within 1-2 points under the experimental data while baseline predictions are 4 points above thye experimental data.


EFFECTS OF THE NUMERICAL VISCOSITY ON THE PERFORMANCE OF THE UH-60A ROTOR

GRID SIZE 149x89x61(808K)

0.005

0.007

0.009

0.011

0.07 0.095 0.12

CT/

CQ/

ExperimentBASELINE (MUSCL3+BL)STVD6+WENO5+BLSTVD8+WENO5+BL

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.05 0.075 0.1

CT/

FM

ExperimentBASELINE (MUSCL3+BL)STVD6+WENO5+BLSTVD8+WENO5+BL

• High order algoritms with very large stencils are quite sensitive to grid spacing, the truncation error can increase compared to 6th an lower order schemes on coarse grid.


EFFECTS OF THE HIGHER ORDER METRICS ON THE PERFORMANCE OF THE UH-60A ROTOR

GRID SIZE 149x89x61(808K)

0.005

0.007

0.009

0.011

0.07 0.095 0.12

CT/

CQ/

ExperimentBASELINE (MUSCL3+MET2+BL)STVD6+MUSCL3+MET2+BLSTVD6+MUSCL3+MET4+BL

0.65

0.67

0.69

0.71

0.73

0.75

0.77

0.79

0.81

0.05 0.075 0.1

CT/

FM

ExperimentBASELINE (MUSCL3+MET2+BL)STVD6+MUSCL3+MET2+BLSTVD6+MUSCL3+MET4+BL

• STVD6+MUSCL3+MET4+BL shows an improvement in predictions.•This was not realized with STVD8.


EFFECTS OF THE HIGHER ORDER LOAD INTEGRATIONS ON THE PERFORMANCE OF

THE UH-60A ROTOR GRID SIZE 149x89x61(808K)

0.0055

0.0075

0.0095

0.0115

0.07 0.095 0.12

CT/

CQ/

ExperimentBASELINE (MUSCL3+MET2+INT2+BL)STVD6+MUSCL3+MET4+INT2+BLSTVD6+MUSCL3+MET4+INT4+BL

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.05 0.075 0.1

CT/

FM

ExperimentBASELINE (MUSCL3+MET2+INT2+BL)STVD6+MUSCL3+MET4+INT2+BLSTVD6+MUSCL3+MET4+INT4+BL

• The order of load integration does not seem to affect the results except to a very small extent at the highest calculated blade loading.


EFFECTS OF THE TURBULENCE MODEL ON THE PERFORMANCE OF THE UH-60A ROTOR GRID SIZE

149x89x61(808K)

0.005

0.007

0.009

0.011

0.07 0.095 0.12

CT/

CQ/

ExperimentBASELINE (MUSCL3+BL)STVD8+MUSCL3+SASTVD8+MUSCL3+BL

0.65

0.67

0.69

0.71

0.73

0.75

0.77

0.05 0.075 0.1

CT/

FM

ExperimentBASELINE (MUSCL3+BL)STVD8+MUSCL3+SASTVD8+MUSCL3+BL

• SA model requires more power than BL model predictions.•SA tends to give slightly higher skin friction values.•The profile power consumption by the rotor is higher with SA model.


CONVERGENGE HISTORY & CPU TIME

SCHEME GRID SIZE CPU(SGIOctane) Memory CPU Time/step TURNS 149x89x61(808K) R5000 – 200MHz 320 MB 95 seconds TURNS-STVD6-MUSCL3 149x89x61(808K) R5000 – 200MHz 320 MB 99 seconds TURNS-STVD6-WENO5 149x89x61(808K) R5000 – 200MHz 320 MB 110 seconds


CONCLUSIONS-I

• As the formal accuracy of the symmetric schemes increases, the dispersion errors in the solution decreases.

• A high order algorithm can require 1/2 to 1/4 of the points that a lower order algorithm will require in a one-dimensional problem.

• In three dimensions, conservatively 1/2 to 1/8 the number of nodes will be needed with a high order scheme.

• Thus significant savings in the computational resources can be achieved.


CONCLUSIONS-I (continued)

• L2-Norm of the error is computed for each symmetric scheme for the grid study. It is seen that the higher order schemes, have consistently lower errors on a given grid.

• The slope of the curves ( a measure of the order of accuracy of the scheme) is steeper for the higher order schemes compared to low order methods. This means an acceptable level of accuracy can be achieved sooner, if the grid is systematically refined.


CONCLUSIONS-II

• High order schemes have consistently lower dissipative errors compared to low order schemes. They also have fewer and weaker spurious precursor and post-cursor waves indicating that the dispersion errors are small.

• The hover performance calculations on highly stretched grids show that a point of diminishing returns is reached as increasingly higher order algorithms are used. Sixth order algorithms are decidedly better than fourth and third order upwind algorithms.

• On the other hand, eighth order accurate algorithms tend to be worse than sixth order algorithms, due to the very large stencil, and relatively larger truncation errors on stretched grids.


CONCLUSIONS-III

NUMERICAL VISCOSITY ANALYSIS:

• In applications such as BVI interactions and rotor-airframe interactions where vortex capturing is important, the numerical viscosity in the scheme can play an equally important role.

• The present calculations showed that spatially fifth order WENO schemes are a significant improvement over the third order MUSCL numerical viscosity schemes.


CONCLUSIONS-IVHIGH ORDER METRICS ANALYSIS:

• The studies on the effects of high order metrics on the solution was inconclusive, perhaps due to the nonlinear interplay between the metrics calculation and the solution algorithm.

• High order metrics and high order large-stencil algorithms both have large truncation errors on stretched grids, and the combination may perform poorly than their low order counterparts.

• In the calculations studied here, STVD6 algorithms in conjunction with fourth order metrics were found to give the best agreement with experiments.


CONCLUSIONS-V

TURBULENCE MODEL ANALYSIS:

• The one-equation turbulence Spalart-Allmaras model was robust, efficient, and eliminated some of the uncertainties in modeling turbulence effects in the wake, and in estimating mixing length distances.

• In the calculations done here, however, the total power consumption was over-predicted with the SA model, compared to the Baldwin-Lomax model. Until further skin friction data and profile power data for validating turbulence models become available, it is recommended that the baseline Baldwin-Lomax model be used in the calculations.


CONCLUSIONS - VI

• The grid quality can play as big a role in the quality and accuracy of the results as any of the algorithmic and modeling issues considered above.

• When a C-H type of grid designed to handle the periodicity in the solution is used, the grid lines become highly skewed and distorted. Grids can also be skewed in the tip region.

A good quality grid that is only moderately stretched is essential, if the higher order algorithms developed in this study are to prove successful.


RECOMMENDATIONS-I

• High order methods work well on uniform meshes, but sometimes behave unpredictably as the grid is refined, due to the nonlinear interaction between the algorithm and grid qualities such as skewness and strectching. Methods, such as discontinuous Galerkin methods and spectral element, that use a compact stencil must be investigated.

• High order schemes are not always compatible with chimera methods because they require increased grid overlap. They do not work well with distributed computing methods due to the increased data transfer near the zone boundaries. Work is needed on high order algorithms of uniformly high accuracy in the interior and the boundaries.


RECOMMENDATIONS-II• The far-field boundary condition in existing hover

analyses are primitive, and do not ensure global balance of momentum and energy. This may affect wake descent and contraction rate. Additional work is needed to improve the boundary condition treatment.

• Although the higher order methods presented here give good performance data, tip loads still not satisfactory due to the highly stretched/clustered grids near the tip, and the discontinuity in the H-grid slope at the tip.

• Work must be done to improve performance of high order algorithms on highly stretched grids.This is critical to rotor tip design.


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