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
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
Blade Motion
⋆ Johnson, W., “Helicopter Theory”, 1980.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
Articulated Rotor
⋆ Johnson, W., “Helicopter Theory”, 1980.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
Forward Flight
⋆ Johnson, W., “Helicopter Theory”, 1980.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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!
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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 . . .
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
What is the Time Spectral Method?
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
What Has Been Done?
Fully nonlinear methods:1 Harmonic Balance method of Hall, Thomas & Clark (2002):
originally for turbomachinery.
Ekici & Hall (2008): Rotorcraft simultion.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
What Has Been Done?
Fully nonlinear methods:1 Harmonic Balance method of Hall, Thomas & Clark (2002):
originally for turbomachinery.
Ekici & Hall (2008): Rotorcraft simultion.
2 Nonlinear frequency domain (NLFD) of McMullen, Jameson& Alonso (2001, 2002).
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Helicopter SimulationTime Spectral Method
What Has Been Done?
Fully nonlinear methods:1 Harmonic Balance method of Hall, Thomas & Clark (2002):
originally for turbomachinery.
Ekici & Hall (2008): Rotorcraft simultion.
2 Nonlinear frequency domain (NLFD) of McMullen, Jameson& Alonso (2001, 2002).
3 Time Spectral method of Gopinath & Jameson (2005).
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
Fourier Based Time Integration Method
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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 .
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
Nonlinear Frequency Domain
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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
.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
Nonlinear Frequency Domain Method
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)
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
Nonlinear Frequency Domain Method
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
Time Spectral
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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)
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Spectral DerivativeNLFD MethodTime Spectral MethodFourier Collocation Matrix
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Rotorcraft Simulation Results
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Hover Calculations
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Caradonna & Tung Experiment (1981)
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Mesh
(a) View of blade (b) Isometric view
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Case I
Flow Condition:
Mtip = 0.52θc = 0
Ω = 1500 RPM
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Case II
Flow Condition:
Mtip = 0.439θc = 8
Ω = 1250 RPM
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – SLIP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Case III
Flow Condition:
Mtip = 0.877θc = 8
Ω = 2500 RPM
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – SLIP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – SLIP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Euler Calculationwith Time Spectral method
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – CUSP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
What if the Coefficient of Thrustis Not Known in Advance?
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Increase the top and bottom distance
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Increase the top and bottom distance
And run your code for a long, long time . . .
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Velocity Magnitude
(a) 500 (b) 1,500 (c) 2,250
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Velocity Magnitude
(d) 3,000 (e) 3,500 (f) 4,000
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Forward Flight Calculations
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – CUSP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – CUSP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Euler & RANS Calculationswith Four Time Instances (N = 4)
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – CUSP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Time Spectral method used 500 multigrid cycles with 12 timeinstances
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Time Spectral method used 500 multigrid cycles with 12 timeinstances
In terms of the number of multigrid cycles required ...
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Time Spectral method used 500 multigrid cycles with 12 timeinstances
In terms of the number of multigrid cycles required ...
Time Spectral method is 87 times faster
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Time Spectral method used 500 multigrid cycles with 12 timeinstances
In terms of the number of multigrid cycles required ...
Time Spectral method is 87 times faster
In terms of CPU hours ...
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Time Spectral method used 500 multigrid cycles with 12 timeinstances
In terms of the number of multigrid cycles required ...
Time Spectral method is 87 times faster
In terms of CPU hours ...
Time Spectral method is still 7.2 times faster
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Time-Lagged Periodic Boundary Condition
@@
@@
@@
@
@@
@@
@@
@
rr
z
ab
?U∞
?Ω
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Euler Calculations
Collective pitch, θc = 0
Tip Mach number, Mtip = 0.7634
Advance ratio, µ = 0.25
128 × 48 × 32 mesh cells
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
experiment, — JST scheme, – – CUSP scheme
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Lifting Rotor in Forward Flight
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Test Case
Caradonna & Tung rotor
Collective pitch, θc = 8
Tip Mach number, Mtip = 0.7
Advance ratio, µ = 0.2857
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Azimuthal angle, ψ in °
CL
(b) 160× 48× 48
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(c) 192 × 64× 48
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(d) 160×48×48⋆
⋆ with 18 time instances
— Allen, • computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
CL comparison – CUSP 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
Azimuthal angle, ψ in °
CL
(b) 160× 48× 48
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(c) 192 × 64× 48
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(d) 160×48×48⋆
⋆ with 18 time instances
— Allen, • computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
Cp Comparison
Comparison is made at blade section r/R = 0.90
Strong transonic flow on the advancing side
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
Flow SolverHover CalculationsBasic Forward Flight CalculationsComputational CostLifting Forward Flight Calculations
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Confinement
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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||.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Making ǫ Dimensionless and Dynamic
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Making ǫ Dimensionless and Dynamic
Fedkiw et al. (2001) for incompressible Euler equations onstructured meshes
ǫh ∝ ǫh
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Making ǫ Dimensionless and Dynamic
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||
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Robinson (2004)
Chose to scale ǫ is with |u|
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Robinson (2004)
Chose to scale ǫ is with |u|
Factor out |~ω| from s(= ∇|~ω|
|∇|~ω|| × ~ω)
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Robinson (2004)
Chose to scale ǫ is with |u|
Factor out |~ω| from s(= ∇|~ω|
|∇|~ω|| × ~ω)
⇒ |u| · |~ω|
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Robinson (2004)
Chose to scale ǫ is with |u|
Factor out |~ω| from s(= ∇|~ω|
|∇|~ω|| × ~ω)
⇒ |u| · |~ω|
|u · ~ω| ≡ helicity
s = ρ |u · ~ω|
∇ |~ω|
|∇ |~ω||×
~ω
|~ω|
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
New Formulation
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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||.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
NACA 0012 Wing
Test Case:
Euler calculation
Untwisted, untapered wing with NACA 0012 cross section
Aspect ratio of 3α = 5
M∞ = 0.8
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
Figure: ǫ = 0
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
Figure: ǫ = 0.025
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
Figure: ǫ = 0.05
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
Figure: ǫ = 0.075
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Application to Lifting Rotorin Forward Flight
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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) ǫ = 0
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(b) ǫ = 0.05
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(c) ǫ = 0.1
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(d) ǫ = 0.15
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(e) ǫ = 0.2
500 600 700 800 900 1000 11000
0.1
0.2
0.3
0.4
0.5
Azimuthal angle, ψ in °
CL
(f) ǫ = 0.25
— Allen, • computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
x = 2 and x = 3.5
1st time instance, i.e. ψ = 90
(a) ǫ = 0 (b) ǫ = 0.2
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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
× Allen, — computed result
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
x = −1.5
7th time instance, i.e. ψ = 270
(a) ǫ = 0 (b) ǫ = 0.2
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Vorticity Magnitude
x = −3
7th time instance, i.e. ψ = 270
(a) ǫ = 0 (b) ǫ = 0.2
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Vorticity confinement works well for fixed-wing calculations.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Vorticity confinement works well for fixed-wing calculations.
The vortical structure in lifting rotor in forward flight could becontrolled such that the effect of blade–vortex interactionbecame more apparent as ǫ increased.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Vorticity confinement works well for fixed-wing calculations.
The vortical structure in lifting rotor in forward flight could becontrolled such that the effect of blade–vortex interactionbecame more apparent as ǫ increased.
... but further studies are needed for rotorcraft application, atleast with the current mesh geometry.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
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.
Vorticity confinement works well for fixed-wing calculations.
The vortical structure in lifting rotor in forward flight could becontrolled such that the effect of blade–vortex interactionbecame more apparent as ǫ increased.
... 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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).
RANS calculations for nonlifting rotor in forward flight took only 5 hourson four dual-core processors with 500 multigrid cycles.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).
RANS calculations for nonlifting rotor in forward flight took only 5 hourson four dual-core processors with 500 multigrid cycles.
Using the time-lagged boundary condition, computational expense can bereduced by Nb times.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).
RANS calculations for nonlifting rotor in forward flight took only 5 hourson four dual-core processors with 500 multigrid cycles.
Using the time-lagged boundary condition, computational expense can bereduced by Nb times.
New formulation for vorticity confinement has no effect on thedistribution of Cp for fixed-wing transonic flow calculations.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Conclusion
Hover calculation takes much longer than forward flight calculation(surprisingly).
Time Spectral method is approximately 10 times faster than thetraditional backward difference formula (depending on the number oftime instances required).
RANS calculations for nonlifting rotor in forward flight took only 5 hourson four dual-core processors with 500 multigrid cycles.
Using the time-lagged boundary condition, computational expense can bereduced by Nb times.
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%.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
Professor Van Roy
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
Professor Van Roy
Dr. Hahn
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
Professor Van Roy
Dr. Hahn
Dr. Allen
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
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.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
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.
Edwin for bringing “gorilla” back to life a few times last year.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Professor Lele
Professor MacCormack
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.
Edwin for bringing “gorilla” back to life a few times last year.
(I had a few heart attacks last year because of this).
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
Family
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
FamilySisters
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
FamilySistersMum
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
FamilySistersMumDad
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
FamilySistersMumDad
Professor Jameson
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow
OutlineIntroduction
Time Spectral MethodRotorcraft Simulation Results
Vorticity Confinement
IntroductionFormulationCompressible Euler CalculationsApplication to Rotorcraft Flows
Acknowledgements
Friends:Anup, Dave, Jaehoon, Jae-Mo, Minsuk, Sekwan, Sandipan,Tomo.Alvin, Jaiyoung, Norm, Pamornpol, Tonkid, Yongjin.
Ralph, Lynn, Liza, Robin
FamilySistersMumDad
Professor Jameson
Nawee Butsuntorn Time Spectral Method for Rotorcraft Flow