time spectral method for rotorcraft flow with vorticity...

Time Spectral Method for Rotorcraft Flowwith Vorticity Confinement

Nawee ButsuntornDepartment of Mechanical Engineering

Advisor: Antony Jameson

University Oral ExminationMarch 17, 2008

Outline

1 IntroductionHelicopter SimulationTime Spectral Method

2 Time Spectral MethodSpectral DerivativeNLFD Method FormulationTime Spectral Method Formulation

3 Rotorcraft Simulation ResultsHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations

4 Vorticity ConfinementIntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows

OutlineIntroduction

Time Spectral MethodRotorcraft Simulation Results

Vorticity Confinement

Helicopter SimulationTime Spectral Method

Introduction

Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow

OutlineIntroduction




Introduction

Helicopter simulation is very complex and computationallyexpensive:

The flow is highly nonlinear.

Interactions between the vortices with the blades and fuselage.

There is a wide range of scales.


OutlineIntroduction




Introduction

Helicopter simulation is very complex and computationallyexpensive:

The flow is highly nonlinear.

Interactions between the vortices with the blades and fuselage.

There is a wide range of scales.

Blades are highly elastic.

Variety of blade motion:

LeadLagFlappingCollective pitch, cyclic pitch, yaw


OutlineIntroduction




Blade Motion

⋆ Johnson, W., “Helicopter Theory”, 1980.


OutlineIntroduction




Articulated Rotor



OutlineIntroduction




Forward Flight



OutlineIntroduction




Background

A lot has been done over the past 3 decades.

Potential flow calculations:

Caradonna & Isom (1972, 1976)Caradonna & Philippe (1976)Arieli, Taubert & Caughey (1986): the first three-dimensional,full potential flow based on Jameson & Caughey’s FLO22


OutlineIntroduction




Background

A lot has been done over the past 3 decades.

Potential flow calculations:

Caradonna & Isom (1972, 1976)Caradonna & Philippe (1976)Arieli, Taubert & Caughey (1986): the first three-dimensional,full potential flow based on Jameson & Caughey’s FLO22

Euler and Reynolds averaged Navier–Stokes (RANS)calculations:

Agarwal & Deese (1987, 1988)Srinivasan et al. (1991, 1992)Pomin & Wagner (2002, 2004)Allen (2003, 2004, 2005, 2006, 2007): 32 million mesh pointsand 25,000 CPU hours for Euler calculation of a four-bladedrotor!


OutlineIntroduction




Hybrid solvers:

Hassan, Tung & Sankar (1992): vortex wake flow field usingBiot–Savart lawYang et al. (2000, 2002): RANS/Full potential solver coupledwith free/prescribed wake modelBhagwat, Moulton & Caradonna (2005, 2006): RANS withwake model


OutlineIntroduction




Hybrid solvers:

Hassan, Tung & Sankar (1992): vortex wake flow field usingBiot–Savart lawYang et al. (2000, 2002): RANS/Full potential solver coupledwith free/prescribed wake modelBhagwat, Moulton & Caradonna (2005, 2006): RANS withwake model

And numerous countless others . . .


OutlineIntroduction




What is the Time Spectral Method?


OutlineIntroduction




Time Spectral Method

Time integration method based on Fourier representation.

Efficient and accurate method for periodic problems.

No need to Fourier transform variables back and forthbetween time and frequency domains, everything is solved inthe time domain.

Algorithm is easily adapted to the current solvers.

Existing convergence acceleration techniques are applicable.

The method is able to achieve spectral accuracy in theory.


OutlineIntroduction




What Has Been Done?

Fully nonlinear methods:1 Harmonic Balance method of Hall, Thomas & Clark (2002):

originally for turbomachinery.

Ekici & Hall (2008): Rotorcraft simultion.


OutlineIntroduction




What Has Been Done?




2 Nonlinear frequency domain (NLFD) of McMullen, Jameson& Alonso (2001, 2002).


OutlineIntroduction




What Has Been Done?




2 Nonlinear frequency domain (NLFD) of McMullen, Jameson& Alonso (2001, 2002).

3 Time Spectral method of Gopinath & Jameson (2005).


OutlineIntroduction



Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix

Fourier Based Time Integration Method


OutlineIntroduction




Spectral Derivative

Real, periodic function f (x) defined on N equally spaced gridpoints, xj = j∆x where j = 0, 1, 2, . . . ,N − 1. The discrete Fouriertransform and the inverse Fourier transform of f are

fk =1

N

N−1∑

j=0

fje−ikxj , f =

N2−1∑

k=−N2

fkeikxj ,

and the spectral derivative of f is Dfk = ikfk , i.e.

df

dx

∣∣∣∣j

=

N2−1∑

k=−N2+1

Dfkeikxj .


OutlineIntroduction




Nonlinear Frequency Domain


OutlineIntroduction




Nonlinear Frequency Domain

Starting with the governing equations in semi-discrete form:

d

dt(V w) + R(w) = 0,

where

w =

ρρu1

ρu2

ρu3

ρE

.


OutlineIntroduction




Nonlinear Frequency Domain Method

Assume that both w and R(w) are periodic in time. These can berepresented as finite Fourier series.

w =

N2−1∑

k=−N2

wkeikt

and

R(w) =

N2−1∑

k=−N2

Rk(w)eikt


OutlineIntroduction





Seperate equation for each Fourier wave number k, and multiplythe time derivative by ik

ikV wk + Rk(w) = 0.

1 Assume that wk is known at all time instances (this is theinitial condition)

2 Inverse Fourier transform wk back to the physical space toobtain w

3 Calculate the residual R(w)

4 Fourier transform R(w) back to the frequency space to obtain

Rk(w)


OutlineIntroduction





Group Rk(w) + ikV wk together,

Rk(w) + ikV wk = Ik(w).

Introduce pseudo time τ ,

Vdwk

dτ+ Ik(w) = 0. (1)

Solve (1) as a steady state problem using well known convergenceacceleration methods, e.g. modified 5-stage Runge–Kutta timestepping scheme, multigrid, local time stepping, etc.


OutlineIntroduction




Time Spectral


OutlineIntroduction




Time Spectral Method

The discrete Fourier transform of the flow variables w for a timeperiod T is

wk =1

N

N−1∑

n=0

wne−ik 2π

Tn∆t ,

and its inverse transform:

wn =

N2−1∑

k=−N2

wkeik 2π

Tn∆t . (2)


OutlineIntroduction




The spectral derivative of equation (2) with respect to time at then-th time instance is given by

Dwn =2π

T

N2−1∑

k=−N2+1

ikwkeik 2π

Tn∆t .

The right hand side can be written in terms of the flow variableswn as follows:

Dwn =

N−1∑

j=0

d jnw

j

where

d jn =

2πT

12(−1)n−j cot

π(n−j)

N

: n 6= j

0 : n = j.


OutlineIntroduction




Let n − j = −m, one can rewrite the time derivative as

Dwn =

N2−1∑

m=−N2+1

dmw(n+m)

where dm is given by

dm =

2πT

12 (−1)m+1 cot

πmN

: m 6= 0

0 : m = 0.


OutlineIntroduction




The original flow equations in semi-discrete form:

Vdwn

dt+ R(wn) = 0,

becomesVDwn + R(wn) = 0. (3)

Introducing a pseudo time derivative term to equation (3), theequations can now be marched towards a periodic steady stateusing well known convergence acceleration techniques.

Vdwn

dτ+ VDwn + R(wn) = 0.


OutlineIntroduction



Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations

Rotorcraft Simulation Results


OutlineIntroduction




Flow Solver

1 Modified 5-stage Runge–Kutta⋆

2 Local time stepping⋆

3 Multigrid⋆

4 Artificial dissipation schemes:⋆

Jameson–Schmidt–Turkel (JST)Symmetric LImited Positive (SLIP)Convective Upwind and Split Pressure (CUSP)

5 Internal mesh generator via conformal mapping

6 Baldwin–Lomax turbulence model (Baldwin–Lomax, 1978)

⋆ A. Jameson, A perspective on computational algorithms for aerodynamics analysis and design,Progress in Aerospace Sciences, 37, pp. 197–243, 2001.

⋆ A. Jameson, Aerodynamics, Encyclopedia of Computational Mechanics, Ch. 11, 2004.


OutlineIntroduction




Hover Calculations


OutlineIntroduction




Caradonna & Tung Experiment (1981)

Experimental setup:

Untapered, untwisted two-bladed rotor

NACA 0012 section

Aspect ratio of 6

Diameter of the rotor is 2.286 m


OutlineIntroduction




Caradonna & Tung Experiment (1981)


OutlineIntroduction




Mesh

O–H mesh for Euler calculation

Single block with one sector

128 × 48 × 32 cells, 16 cells on the blade

Bottom and top far-field distance is 1.5R from the rotor

Far-field in the spanwise direction is 1.5R from the rotor


OutlineIntroduction




Mesh

(a) View of blade (b) Isometric view


OutlineIntroduction




Boundary Conditions

Top: Riemann invariants

Hub: flow tangency (Euler calculation)

Far-field (spanwise): Riemann invariants

Periodic boundary condition for the periodic boundaries

Bottom:

Nonlifting: Riemann invariantsLifting: one-dimensional momentum theory


OutlineIntroduction




Steady State Periodic Problem

By adding a forcing term (Holmes & Tong, 1984), the equationscan now be solved as a steady state problem.

∫

Ω

∂w

∂tdV +

∮

∂Ωfj · ~n dS =

∫

ΩTdV

where T is defined as

T =

0ρΩu3

0−ρΩu1

0

.


OutlineIntroduction




Case I

Flow Condition:

Mtip = 0.52θc = 0

Ω = 1500 RPM


OutlineIntroduction




Nonlifting Rotor (1)

0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x/c

−C

p

(a) r/R = 0.68

0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x/c

−C

p

(b) r/R = 0.80

experiment, — JST scheme


OutlineIntroduction




Nonlifting Rotor (2)

0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x/c

−C

p

(c) r/R = 0.89

0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x/c

−C

p

(d) r/R = 0.96



OutlineIntroduction




Case II

Flow Condition:

Mtip = 0.439θc = 8

Ω = 1250 RPM


OutlineIntroduction




Subsonic Lifting Rotor (1)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(a) r/R = 0.68

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(b) r/R = 0.80

experiment, — JST scheme, – – SLIP scheme


OutlineIntroduction




Subsonic Lifting Rotor (2)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(c) r/R = 0.89

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(d) r/R = 0.96



OutlineIntroduction




Case III

Flow Condition:

Mtip = 0.877θc = 8

Ω = 2500 RPM


OutlineIntroduction




Transonic Lifting Rotor (1)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(a) r/R = 0.68

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(b) r/R = 0.80



OutlineIntroduction




Transonic Lifting Rotor (2)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(c) r/R = 0.89

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(d) r/R = 0.96



OutlineIntroduction




Euler Calculationwith Time Spectral method


OutlineIntroduction




Transonic Lifting Rotor: Time Spectral Method (1)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(a) r/R = 0.68

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(b) r/R = 0.80

experiment, — JST scheme, – – CUSP scheme


OutlineIntroduction




Transonic Lifting Rotor: Time Spectral Method (2)

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(c) r/R = 0.89

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(d) r/R = 0.96



OutlineIntroduction




Calculations were run on four dual-core processors (clockspeed is 3.0 GHz)

250 multigrid cycles

The averaged residual reduced by almost four orders ofmagnitude


OutlineIntroduction




Calculations were run on four dual-core processors (clockspeed is 3.0 GHz)

250 multigrid cycles

The averaged residual reduced by almost four orders ofmagnitude

13 minutes


OutlineIntroduction




What if the Coefficient of Thrustis Not Known in Advance?


OutlineIntroduction




Increase the top and bottom distance


OutlineIntroduction




Increase the top and bottom distance

And run your code for a long, long time . . .


OutlineIntroduction




Velocity Magnitude

(a) 500 (b) 1,500 (c) 2,250


OutlineIntroduction




Velocity Magnitude

(d) 3,000 (e) 3,500 (f) 4,000


OutlineIntroduction




160 × 64 × 48 Mesh Cells, 3,500 time steps

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(a) r/R = 0.68

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(b) r/R = 0.80



OutlineIntroduction




160 × 64 × 48 Mesh Cells, 3,500 time steps

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(c) r/R = 0.89

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(d) r/R = 0.96



OutlineIntroduction




Forward Flight Calculations


OutlineIntroduction




Caradonna et al. Experiment (1984)

Experimental setup:

Untapered, untwisted two-bladed rotor

NACA 0012 section

Aspect ratio of 7

Diameter of the rotor is 7 ft

Chord is 6 in


OutlineIntroduction




Nonlifting Rotor in Forward Flight

Case IV

Flow Condition:

θc = 0

Mtip = 0.8µ = 0.2Re = 2.89 × 106

Twelve time instances were used, N = 12


OutlineIntroduction




Mesh

Euler: 128 × 48 × 32 cells per blade, 16 cells on the blade.

RANS: 192 × 64 × 48 cells per blade, 32 cells on the blade.

(a) Isometric view (b) Top view


OutlineIntroduction




Comparison with the Experimental Data

Dissipation schemes are JST and CUSP

Results are compared at six azimuthal angles on theadvancing side:

(a) ψ = 30

(b) ψ = 60

(c) ψ = 90

(d) ψ = 120

(e) ψ = 150

(f) ψ = 180


OutlineIntroduction




Euler Calculations

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




RANS Calculations

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




Euler & RANS Calculationswith Four Time Instances (N = 4)


OutlineIntroduction




Euler/RANS Calculations with four time instances

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(a) ψ = 90

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(b) ψ = 180

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

x/c

−C

p

(d) ψ = 180



OutlineIntroduction




Computational Cost

300 multigrid cycles for Euler calculations.

Residual reduced by four orders of magnitude.

500 multigrid cycles for RANS calculations.

5 hours on four dual-core processors (clock speed is 3.0 GHz).Residual reduced by three orders of magnitude.


OutlineIntroduction




Comparison with Backward Difference Formula (BDF)⋆

V

3

2∆twn+1 −

4

2∆twn +

1

2∆twn−1

+ R

(wn+1

)= 0.

Periodicity is established, not enforced.

Usually requires at least 4 cycles (for pitching airfoil/wing).

⋆ A. Jameson, “Time Dependent Calculations Using Multigrid, with Applications to

Unsteady Flows Past Airfoils and Wings”, AIAA Paper 1991–1596.


OutlineIntroduction




Comparison with Backward Difference Formula (BDF)

For the same RANS calculations, the BDF would need:

180 time steps per revolution

40 multigrid cycles per time step

6 cycles to convergence

⇒ 43200 steps


OutlineIntroduction









⇒ 43200 steps

Time Spectral method used 500 multigrid cycles with 12 timeinstances


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⇒ 43200 steps


In terms of the number of multigrid cycles required ...


OutlineIntroduction









⇒ 43200 steps



Time Spectral method is 87 times faster


OutlineIntroduction









⇒ 43200 steps




In terms of CPU hours ...


OutlineIntroduction









⇒ 43200 steps




In terms of CPU hours ...

Time Spectral method is still 7.2 times faster


OutlineIntroduction




Time-Lagged Periodic Boundary Condition

First proposed by Ekici & Hall⋆ (2008)

One blade is required for forward flight simultions

Further saving of Nb times

where Nb is the number of blades per rotor

w(r , ψ, z , t) = w

(r , ψ −

2π

N, z , t −

T

N

)

⋆ Ekici, Hall & Dowell, “Computationally Fast Harmonic Balance Methods for

Unsteady Aerodynamic Predictions of Helicopter Rotors”, AIAA Paper 2008–1439.


OutlineIntroduction




Time-Lagged Periodic Boundary Condition

@@

@@

@@

@

@@

@@

@@

@

rr

z

ab

?U∞

?Ω


OutlineIntroduction




Euler Calculations

Collective pitch, θc = 0

Tip Mach number, Mtip = 0.7634

Advance ratio, µ = 0.25

128 × 48 × 32 mesh cells


OutlineIntroduction




Euler Calculations

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(b) ψ = 60

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




Lifting Rotor in Forward Flight


OutlineIntroduction




Test Case

Caradonna & Tung rotor

Collective pitch, θc = 8

Tip Mach number, Mtip = 0.7

Advance ratio, µ = 0.2857


OutlineIntroduction




Numerical Data Provided by C. B. Allen

Over 2 million mesh points around the two blades and hub(not including the other blocks that cover the far-fields)

BDF time stepping scheme

180 steps per revolution

6 revolutions

Periodicity is established after the second revolution

JST dissipation scheme

70 3-level V-cycle multigrid cycles per time step

⋆ C.B. Allen, “An Unsteady Multiblock Multigrid Scheme for Lifting Forward Flight

Rotor Simulation”, International Journal for Numerical Methods in Fluids, 2004.


OutlineIntroduction




Lift Comparison

Load variation on each blade around the azimuth

CL =Fy

12ρ (ΩR)2 c R

whereFy = force in the y direction

Ω = angular velocity

c = chord

R = rotor radius

ρ = density


OutlineIntroduction




CL comparison – JST Scheme

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5

Azimuthal angle, ψ in °

CL

(a) 128× 48× 32

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(b) 160× 48× 48

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(c) 192 × 64× 48

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(d) 160×48×48⋆

⋆ with 18 time instances

— Allen, • computed result


OutlineIntroduction




CL comparison – CUSP Scheme

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(a) 128× 48× 32

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(b) 160× 48× 48

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(c) 192 × 64× 48

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(d) 160×48×48⋆

⋆ with 18 time instances



OutlineIntroduction




Cp Comparison

Comparison is made at blade section r/R = 0.90

Strong transonic flow on the advancing side


OutlineIntroduction




160 × 48 × 48: JST scheme (Advancing Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(f) ψ = 180

× Allen, — computed result


OutlineIntroduction




160 × 48 × 48: JST scheme (Retreating Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(g) ψ = 210

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p(h) ψ = 240

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(i) ψ = .250

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(j) ψ = 300

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(k) ψ = 330

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(l) ψ = 360



OutlineIntroduction




160 × 48 × 48: CUSP scheme (Advancing Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




160 × 48 × 48: CUSP scheme (Retreating Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(g) ψ = 210

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p(h) ψ = 240

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x/c

−C

p

(i) ψ = .250

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(j) ψ = 300

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(k) ψ = 330

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(l) ψ = 360



OutlineIntroduction



IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows



OutlineIntroduction




What is Vorticity Confinement?

John Steinhoff first suggested the idea in 1994.

A forcing term added to the momentum equations (inviscid,incompressible), “so that as the vorticity diffuses away fromthe centroids of vortical regions, it is transported back”.

Vorticity is added in the direction normal to both ~ω and thegradient |~ω|.

Unfortunately momentum is not conserved.


OutlineIntroduction




Original Formulation

Steinhoff & Underhill (1994); Steinhoff (1994):

∂u

∂t+ (u · ∇) u =

1

ρ∇p + µ∇2u − ǫs

where the simplest form of s is

s =∇|~ω|

|∇|~ω||× ~ω


OutlineIntroduction




Compressible Formulation

Hu & Grossman (2001); Hu et al. (2001) and Dadone et al.

(2001) introduced a body force per unit mass term to the totalenergy equation:

∫

Ω

∂w

∂tdV +

∮

∂Ωfj · ndS = −

∫

ΩǫsdV

where ~s is now:

~s =

0ρ(n × ~ω) · iρ(n × ~ω) · jρ(n × ~ω) · kρ(n × ~ω) · u

and n =∇|~ω|

|∇|~ω||.


OutlineIntroduction




Making ǫ Dimensionless and Dynamic


OutlineIntroduction





Fedkiw et al. (2001) for incompressible Euler equations onstructured meshes

ǫh ∝ ǫh


OutlineIntroduction





Fedkiw et al. (2001) for incompressible Euler equations onstructured meshes

ǫh ∝ ǫh

Lohner & Yang (2002); Lohner et al. (2002) forincompressible RANS calculations on unstructured meshes

ǫv ∝

ǫ|u|ǫh|~ω|ǫh2|∇|~ω||


OutlineIntroduction




Robinson (2004)

Chose to scale ǫ is with |u|


OutlineIntroduction




Robinson (2004)


Factor out |~ω| from s(= ∇|~ω|

|∇|~ω|| × ~ω)


OutlineIntroduction




Robinson (2004)



|∇|~ω|| × ~ω)

⇒ |u| · |~ω|


OutlineIntroduction




Robinson (2004)



|∇|~ω|| × ~ω)

⇒ |u| · |~ω|

|u · ~ω| ≡ helicity

s = ρ |u · ~ω|

∇ |~ω|

|∇ |~ω||×

~ω

|~ω|


OutlineIntroduction




New Formulation


OutlineIntroduction




New Formulation

s = |u · ~ω|

[1 + log10

(1 +

V

Vaveraged

)1/3]

0

ρ[n × ~ω

|~ω|

]· i

ρ[n × ~ω

|~ω|

]· j

ρ[n × ~ω

|~ω|

]· k

ρ[n × ~ω

|~ω|

]· u

where

n =∇|~ω|

|∇|~ω||.


OutlineIntroduction




NACA 0012 Wing

Test Case:

Euler calculation

Untwisted, untapered wing with NACA 0012 cross section

Aspect ratio of 3α = 5

M∞ = 0.8


OutlineIntroduction




Vorticity Magnitude

Figure: ǫ = 0


OutlineIntroduction




Vorticity Magnitude

Figure: ǫ = 0.025


OutlineIntroduction




Vorticity Magnitude

Figure: ǫ = 0.05


OutlineIntroduction




Vorticity Magnitude

Figure: ǫ = 0.075


OutlineIntroduction




cd and cl at Three Different Spans

z = 0.891 z = 1.828 z = 2.766ǫ cl cd cl cd cl cd

0 0.7098 0.0792 0.6123 0.0651 0.3869 0.03940.025 0.7091 0.0791 0.6114 0.0650 0.3851 0.03930.050 0.7083 0.0790 0.6103 0.0649 0.3833 0.03910.075 0.7074 0.0788 0.6093 0.0647 0.3817 0.0389

0.3% difference in cl and 0.5% difference in cd at z = 0.891

1.3% difference in both cl and cd at z = 2.766


OutlineIntroduction




Cp Plots

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

1.5

CP

x

(a) z = 0.891

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

1.5

CP

x

(b) z = 1.828

0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

1.5

CP

x

(c) z = 2.766

— — ǫ = 0, · · · ǫ = 0.025, – · – ǫ = 0.05, – – ǫ = 0.075


OutlineIntroduction




Application to Lifting Rotorin Forward Flight


OutlineIntroduction




CL Comparison: JST Scheme

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(a) ǫ = 0

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(b) ǫ = 0.05

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(c) ǫ = 0.1

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(d) ǫ = 0.15

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(e) ǫ = 0.2

500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5


CL

(f) ǫ = 0.25



OutlineIntroduction




160 × 48 × 48: JST scheme (Advancing Side), ǫ = 0.2

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




160 × 48 × 48: JST scheme (Advancing Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(a) ψ = 30

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p(b) ψ = 60

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(c) ψ = 90

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

x/c

−C

p

(d) ψ = 120

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(e) ψ = 150

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

x/c

−C

p

(f) ψ = 180



OutlineIntroduction




Vorticity Magnitude

x = 2 and x = 3.5

1st time instance, i.e. ψ = 90

(a) ǫ = 0 (b) ǫ = 0.2


OutlineIntroduction




160 × 48 × 48: JST scheme (Retreating Side), ǫ = 0.2

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(a) ψ = 210

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p(b) ψ = 240

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(c) ψ = 255

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(d) ψ = 300

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(e) ψ = 330

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(f) ψ = 360



OutlineIntroduction




160 × 48 × 48: JST scheme (Retreating Side)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(a) ψ = 210

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p(b) ψ = 240

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(c) ψ = 255

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(d) ψ = 300

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(e) ψ = 330

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x/c

−C

p

(f) ψ = 360



OutlineIntroduction




Vorticity Magnitude

x = −1.5

7th time instance, i.e. ψ = 270

(a) ǫ = 0 (b) ǫ = 0.2


OutlineIntroduction




Vorticity Magnitude

x = −3

7th time instance, i.e. ψ = 270

(a) ǫ = 0 (b) ǫ = 0.2


OutlineIntroduction




Future Work & Summary

Time Spectral method has proved to be an efficient methodfor periodic problems, providing that the number of timeinstances are enough to capture the smallest frequency.


OutlineIntroduction






Vorticity confinement works well for fixed-wing calculations.


OutlineIntroduction







The vortical structure in lifting rotor in forward flight could becontrolled such that the effect of blade–vortex interactionbecame more apparent as ǫ increased.


OutlineIntroduction








... but further studies are needed for rotorcraft application, atleast with the current mesh geometry.


OutlineIntroduction








... but further studies are needed for rotorcraft application, atleast with the current mesh geometry.

Perhaps H-mesh would be better suited, or one can resort tooverset or unstructured meshes.


OutlineIntroduction




Conclusion

Hover calculation takes much longer than forward flight calculation(surprisingly).


OutlineIntroduction




Conclusion


Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).


OutlineIntroduction




Conclusion



RANS calculations for nonlifting rotor in forward flight took only 5 hourson four dual-core processors with 500 multigrid cycles.


OutlineIntroduction




Conclusion




Using the time-lagged boundary condition, computational expense can bereduced by Nb times.


OutlineIntroduction




Conclusion





New formulation for vorticity confinement has no effect on thedistribution of Cp for fixed-wing transonic flow calculations.


OutlineIntroduction




Conclusion





New formulation for vorticity confinement has no effect on thedistribution of Cp for fixed-wing transonic flow calculations.

The maximum error for cl and cd for was only 1.3%.


OutlineIntroduction




Acknowledgements

Professor Lele


OutlineIntroduction




Acknowledgements

Professor Lele

Professor MacCormack


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy

Dr. Hahn


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy

Dr. Hahn

Dr. Allen


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy

Dr. Hahn

Dr. Allen

Lab/office mates

Aaron, Andy, Charlie, Edmond, Jen-Der, Matt, Rui, Sachin &Sean.Arathi, Georg, Kasidit, Karthik, Kihwan, John, Sriram.Britton & Chris.


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy

Dr. Hahn

Dr. Allen

Lab/office mates


Edwin for bringing “gorilla” back to life a few times last year.


OutlineIntroduction




Acknowledgements

Professor Lele


Professor Van Roy

Dr. Hahn

Dr. Allen

Lab/office mates


Edwin for bringing “gorilla” back to life a few times last year.

(I had a few heart attacks last year because of this).


OutlineIntroduction




Acknowledgements

Friends:


OutlineIntroduction




Acknowledgements

Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.


OutlineIntroduction




Acknowledgements

Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.


OutlineIntroduction




Acknowledgements


Ralph, Lynn, Liza, Robin


OutlineIntroduction




Acknowledgements



Family


OutlineIntroduction




Acknowledgements



FamilySisters


OutlineIntroduction




Acknowledgements



FamilySistersMum


OutlineIntroduction




Acknowledgements



FamilySistersMumDad


OutlineIntroduction




Acknowledgements



FamilySistersMumDad

Professor Jameson


OutlineIntroduction




Questions?


time spectral method for rotorcraft flow with vorticity...

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