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Multi-scale Method Development for Turbine Heat Transfer and Aerodynamics

(MUSAF II, Toulouse, Sept 18-20, 2013)

L. He

Rolls-Royce/Royal Academy of Engineering Professor

of Computational Aerothermal Engineering,

Department of Engineering Science, Oxford University

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Background Problem Statement

- scale disparities in turbine aero-thermal problems

Unsteady Conjugate Heat Transfer (in time)

- fluid-solid interfacing (for periodic unsteadiness & LES)

‘Block-spectral’ Method (in space)

- resolving micro scales (film cooling, surface roughness)

OUTLINE

High Pressure Turbine - ‘Core of the Heart’

High Pressure, High Speed

- Pressure ratio ~ 40+

- HP Turbine shaft ~ 10,000 RPM

- Transonic flow (Mach No >1)

High Temperature - high gas temperature (1800K+),

(for high efficiency & work output)

- Blade metal melts at 1200K!

Multi-scales (in time & space) Fluid – fast convection

Solid – slow conduction (tS / tF ~ 103)

Main flow path – blade chord C

Cooling holes ~ 10-2 C

compressor

High Pressure

(H.P.) turbine

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Timescale Disparity in Aero-Heat Transfer Interaction (wall condition for the energy equation if non-adiabatic ?)

- For high-speed flow with high heat transfer: large T & variations in near-wall/wakes (i.e. work load/losses)

erroneous flow-only (adiabatic) solution

(regardless of flow model fidelities!)

- Fluid-solid ‘Conjugate Heat Transfer’ (CHT)

- STEADY: working with domain dependent stepping

- UNSTEADY: limited by conflicting requirements

Maximum Time Step dictated by resolving fast unsteadiness (fluid);

Minimum Time Scale dictated by covering slow conduction (solid).

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Exp(-S/R)

1.4

1.34

1.28

1.22

1.16

1.1

1.04

0.98

0.92

0.86

0.8

0.74

0.68

0.62

0.56

0.5

Hot-streak

P.S.Heating

URANS (He et al 2004)

LES (He 2013)

HP turbine with combustor exit

hot streaks (Khanal et al 2012)

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(to ‘realign’ mismatched time scales)

Fluid: solved in time-domain;

Solid: solved in frequency domain:

Ts=T0s+ Σ Ascos(wt) + Bs sin(wt)

(T0s As , Bs are all time-independent (“steady”)

Unsteady solid domain solved in a Steady manner

Hybrid Time/Frequency-domain Approach:

Implemented in a density based solver (He 2000) 3-D Unsteady Navier-Stokes in multi-block meshed domain;

Hex cell-centred upwind (AUSM) finite volume;

4-stage Runge-Kutta time-marching;

Local time stepping & multi-grid;

Implicit dual timing for unsteady flow;

(solid domain: fluid energy equation with zero velocity)

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Verification of Conduction and Convection

Solution Capabilities of CFD solver

0

1000

2000

3000

4000

5000

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1N

uP.S. S/S0 S.S.

Fully Turbulent

Trpped on S.S.

Exp (P.S)

Exp (S.S.)

Conduction Solution

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1(R-R1)/(R2-R1)

(T-T

1)/

(T2

-T1

)

Calc (FV CFD)

Analytical

Blade Surface HTC

0

5

10

15

20

0 0.5 1X/L

NUx

Calc

Analytic

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Semi-Analytical Unsteady Interface Method (He and Oldfield, 2009)

Basic assumption: locally 1D semi-infinite solid domain (valid for high frequencies with small ‘penetration depth’ in solid)

Analytical link in harmonics

between heat flux (qw) and temperature (Tw ):

(TFTq – “transfer function” )

Harmonic Balancing (heat flux / temperature continuity):

(n = 0,1,2, ……Nh)

Discrete for fluid side (FD) & Analytical for solid side:

(n = 0,1,2, ……Nh)

Tf Ts

xf xs

Tw

TBC wTqw TTFq ˆ ˆ

nSnF qq )ˆ( )ˆ(

])( ,)ˆ[(f )ˆ( nTqnfnw TFTT

fluid solid

Impact of Unsteady Surface Temperature

on Heat-flux Prediction (single harmonic)

At 10 Hz , ~33% over-prediction in unsteady heat flux

At 100 Hz, ~18% over-prediction

At 1000 Hz, ~6% over-prediction

Heat Flux Time Trace (HTC=3000 W/m**2/K, f = 10 Hz)

-500000

-250000

0

250000

500000

0 180 360 540 720

Time (wt, deg)

q (

W/m

**2)

q without Twall variation

q with Twall variation

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Conjugate Heat Transfer with LES

Basic Hypothesis: Turbulence at a spatial point ‘deterministically’

manifested by the corresponding Fourier spectrum

Tu Intensity: analogue of ‘Deterministic Stress’ for Nh harmonics

Tu Length-scale: Shape of the spectrum

])u()u( )u()u[(2

1)uu( inin

N

1nrnrn

'' h

1, 2, ………... , Nh

FT on the fly input to the semi-analytical interface condition

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Inflow Turbulence Fluctuations

(Synthetically Randomised Vortices)

Short scale

Long scale

Validation for Inflow Turbulence

Development in Duct (Tu=3.5%)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00

Tu/T

u0

Rex (x105)

Present LES

DNS (Jacobs & Durbin)

LES Conjugate Heat Transfer (Aerothermal Characteristics, Internal Cooled Nozzle, 30 Fourier modes)

Heat Flux (Quasi-steady vs Unsteady Wall BC)

-500

0

500

1000

1500

2000

2500

3000

0 10000 20000 30000 40000

Nu

q

Time step

Unsteady (harmonic-spectral)

Quasi Steady

Shear Stress (Influence of Heat Transfer on Aerodynamics!)

0

100

200

300

400

500

0 10000 20000 30000

Sh

ear

Str

ess

Time Step

Conjugate HT LES

0

100

200

300

400

500

0 10000 20000 30000

Sh

ea

r S

tre

ss

Time Step

Adiabatic LES

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Spatial Length-scale Disparity in Film-Cooling etc

Macro: Blade Chord (passage P & Vel field)

Micro: Film cooling hole (< 1%C),

(mixing of coolant with mainstream)

Compounding Challenge: Large number of cooling holes

( ~102+ )

Direct solutions with all holes resolved are prohibitively costly!

(Oldfield, 2007)

Macro & Micro scales exist in many problems:

- Film/effusion cooling: Cooling hole vs Blade

- Surface treatment/roughness: Dimple vs Blade

Scale-dependent Solvability

Scale-Dependent Solvability Behaviour

- Locally: micro scales of high gradient (needing high resolution)

- Globally: smooth variation among ‘similar’ meso-structures

‘Block-Spectral’ Model (He 2010)

Resolve micro scales (RANS /LES); Avoid solving large domain with very fine mesh.

Set up spectral block-block variations (pointwise) Simultaneous ‘mapping’ to the full domain;

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Pointwise Spectrum for Block Boundary Points

Main (macro) stream only ‘sees’ small

(micro) blocks through boundary faces

For each boundary point (i, j),

variable changes from block to block

(1-D variation wrt block index).

For NB blocks: )(GF)U()(U

BN

1j,i0j,ij,i

i,j

i,j

i,j

i,j

1

2

3

4

For a given basis function G(), only need to solve enough

blocks to determine the coefficients Fi,j

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i,j

i,j

i,j

1

2

3

4

Pointwise Fourier Spectrum (periodic, or “mirroring”/padding for non-periodic)

)]nsin()B()ncos()A[()U(U Lj,inL

N

1nj,in0j,iL,j,i

F

- Fourier Shape of variation with NF harmonics

- for each mesh point (i,j) for Block L :

i,j

- 2NF+1 blocks to be solved to fix the spectrum

Double Fourier Series Shape (M x N):

- (2M+1)(2N+1) blocks to be solved

)]nsin()msin(D)nsin()mcos(C

)ncos()msin(B)ncos()mcos(A[)U(

JIn,mJIn,m

JIn,mJI

hN,hM

0n,0m

n,mn,mJ,I

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Sample 2D Film-cooling Test Cases

Direct (39 holes) Spectral (4 holes)

Hot (T01)

Cold ( 0.5 T01) 0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

X /L

T0

/T

01

Direct (39holes)

FT (10 holes)

FT (7 holes)

FT (4 holes)

1.E-03

1.E-02

1.E-01

1 10 100

L2 Error-Norm

Number of Modes Solved

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Direct Solution

(solving 992 holes)

Block-Spectral

(solving 8 holes)

T01

0.5T01

3D Effusion Cooling (mass & heat transfer via large number of micro holes)

Surface temperature under a distorted infow

Surface Pressure and heat flux (1600 dimples)

(with inlet P0 distortion)

Direct Solution Block Spectral

(Solve 9 dimples)

Non-D Heat Flux (1600 dimples)

0

200

400

600

0 0.2 0.4 0.6 0.8 1

X/L

Nu

Direct Solution (1600 dimples)

Block Spectral ( 21 dimples )

Block Spectral ( 15 dimples )

Block Spectral ( 9 dimples )

Smooth Surface

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(Solve 1600 dimples)

Duct Acoustic Liner for Noise Reduction (Block spectral model of solving 1/30 micro-cavities)

Summary

Two Methods for Multi-scale Problems:

Time scale disparity (fluid-solid heat transfer)

- Harmonic transfer function method:

unified and consistent interfacing condition/framework

for periodic & turbulence unsteady simulations.

Spatial scale disparity (cooling, micro structures)

- Block-spectral method:

macro & micro scales resolved by the same model/numerics

Some Refs: L. He, “Block-spectral Mapping for Multi-scale Solution”,

J. of Computational Physics, Vol.250 (2013).

L. He, “Fourier Spectral Method for Multi-scale Aerothermal Analysis”,

International J. of Computational Fluid Dynamics, Vol.27, No2 (2013).