Grid-free Methods for Compressible Flows Praveen. C ARDB CFD Center - - PowerPoint PPT Presentation

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Fluid Dynamics Colloquium

Grid-free Methods for Compressible Flows

• Praveen. C∗

ARDB CFD Center

• Dept. of Aerospace Engg., IISc

Bangalore-12 Research supervisor: Prof. S M Deshpande

• Dr. A K Ghosh (ADA)

∗CTFD Division, NAL, Bangalore-17

email: cpravn@gmail.com November 9, 2005

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Outline of presentation

• 1. Why meshless methods ?
• 2. Kinetic schemes
• 3. FAME mesh and connectivity generation
• 4. Tests for connectivity
• 5. 3-D results with q-LSKUM on FAME mesh
• 6. Kinetic Meshless Method
• 7. 2-D results with KMM

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Why meshless methods ?

• Grid generation - complex and time-consuming
• 3-D configurations - several hundred man-hours
• Large deformations - moving bodies, extrusion, fragmentation, moving interfaces
• Meshless methods - do not require a mesh - only point distributions and local connectivity
• ✁

Connectivity

C P

• Point distribution

Connectivity

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Why meshless methods ?

• Grid generation - complex and time-consuming
• 3-D configurations - several hundred man-hours
• Large deformations - moving bodies, extrusion, fragmentation
• Meshless methods - do not require a mesh - only point distributions and local connectivity
• 1. Any type of grid or combination of grids (FAME) can be used. Different topologies

can be combined (Ramesh, Anandhanarayanan)

• chimera clouds - few 10s of man-hours (Anandhanarayanan)
• 2. Easier to generate point distributions - fast and automatic point generation
• L¨
• hner et al. - advancing front point generation
• Varma et al. - quadtree/octree point generation
• 3. Simple data structure - ease of adaptation - add points wherever required
• 4. Include a priori knowledge about local behaviour of solution
• 5. Less sensitive to quality of point distributions
• 6. Well suited for moving boundary problems

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Meshless methods Collocation methods Integral methods LSKUM - Ghosh and Deshpande DEM - Nayroles Batina EFG - Belytschko et al. Morinishi PUFEM - Duarte and Oden LSFDU - Balakrishnan et al. hp-clouds - Babuska and Melenk FPM - L¨

• hner et al.

FVPM - Heitel, Junk et al. Simple and easy to implement Requires numerical quadrature Truly meshless Background mesh required Fluid problems Structural mechanics

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Meshless methods Collocation methods Integral methods LSKUM - Ghosh and Deshpande DEM - Nayroles Batina EFG - Belytschko et al. Morinishi PUFEM - Duarte and Oden LSFDU - Balakrishnan et al. hp-clouds - Babuska and Melenk FPM - L¨

• hner et al.

FVPM - Heitel, Junk et al. Simple and easy to implement Requires numerical quadrature Truly meshless Background mesh required Fluid problems Structural mechanics LSKUM - Least Squares Kinetic Upwind Method (1989)

• meshless, kinetic, upwind scheme
• subsonic, transonic, supersonic, hypersonic flows
• higher order scheme (Ghosh and Deshpande, Dauhoo et al.)
• 2-D moving body (Ramesh, Sachin, Chandrashekar)
• 3-D complex configurations (Ramesh, Anandhanarayanan, Mahendra et al.)
• 2-D laminar viscous flows (Mahendra et al., Anandhanarayanan)

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More applications of meshless methods

• Structural mechanics

– Extrusion and molding - large mesh deformation – Casting - propagation of interfaces between solids and liquids – Failure processes - propagation of cracks

• Airbag simulation (Kuhnert et al.)
• fluid-structure interaction
• Simulation of high explosive detonation, blast propagation, and shock wave diffraction

for determining and assessing target vulnerability and weapon lethality (Lohner et al.) – Hybrid approach - Cartesian + gridless – Overall speed-up of 8, and – memory savings of one order compared to unstructured grid method – Accuracy comparable to unstructured grid method

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Basic Idea

• Assume a function variation

fh = fo + a(x − xo) + b(y − yo)

• Fit (a, b) using least squares
• Use fh as a shape function, or
• Estimate derivatives

∂f ∂x ≈ ∂fh ∂x = a, ∂f ∂y ≈ ∂fh ∂y = b

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Full stencil

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Split stencil

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Euler equations

• System of hyperbolic conservation laws - mass, momentum and energy

∂U ∂t + ∂ ∂xG X + ∂ ∂yGY + ∂ ∂zG Z = 0

• Vector of conserved variables

U =       ρ ρux ρuy ρuz E       where ρ = Density ux, uy, uz = Fluid velocity E = Energy per unit volume

• Flux vectors

G X =       ρux p + ρu2

x

ρuxuy ρuxuz (E + p)ux       , GY =       ρuy ρuxuy p + ρu2

y

ρuyuz (E + p)uy       , G Z =       ρuz ρuxuz ρuyuz p + ρu2

z

(E + p)uz      

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Least Squares Kinetic Upwind Method

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Kinetic schemes

• Exploit connection between Boltzmann and Euler/Navier-Stokes equations

∂F ∂t + v∂F ∂x = 0

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Kinetic schemes

• Exploit connection between Boltzmann and Euler/Navier-Stokes equations

∂F ∂t + v∂F ∂x = 0 = ⇒ F(x, t + ∆t) = F(x − v∆t, t)

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Kinetic schemes

• Exploit connection between Boltzmann and Euler/Navier-Stokes equations

∂F ∂t + v∂F ∂x = 0 = ⇒ F(x, t + ∆t) = F(x − v∆t, t)

• Stencil for LSKUM
• Semi-discrete LSKUM approximation

dFo dt + v + |v| 2 ∂F ∂x

• C1
• v≥0

+ v − |v| 2 ∂F ∂x

• C2
• v≤0

= 0

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• Moments of discretized BE

dUo dt + ∂G+ ∂x

• C1

+ ∂G− ∂x

• C2

= 0

• In 3-D we need six split stencils (two per axis)

dUo dt + ∂ ∂xG X+

• ∆x≤0

+ ∂ ∂xG X−

• ∆x≥0

+ ∂ ∂yGY +

• ∆y≤0

+ ∂ ∂yGY −

• ∆y≥0

+ ∂ ∂zG Z+

• ∆z≤0

+ ∂ ∂zG Z−

• ∆z≥0

= 0

• Integrate the ODE system in time using a Runge-Kutta scheme
• Robust scheme

– Entropy consistent – Positivity preservation

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FAME mesh and connectivity generation

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FAME mesh

• FAME - Feature Associated Mesh Embedding (DERA, UK, now called QinetiQ)
• Multiple overlapping meshes - body-fitted mesh around simple parts and + background

Cartesian mesh

• Cartesian mesh adapted based on local features - geometry and body-fitted mesh
• Local body-fitted mesh - 4 to 8 layers in the wall normal direction
• Extra meshes to resolve sharp geometrical features
• wing-fuselage intersection, trailing-edge, wing-tip

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LSKUM + FAME

• Store separation problem - multiple bodies in relative motion
• Moving grids: Refine, coarsen and blank
• Interpolation in overlapping regions
• Grid-free method better suited for this application
• No interpolation required - consistent update at all points
• Collaborative project between QinetiQ, ADA and CFD Center
• Phase I - demonstrate LSKUM on stationary FAME mesh

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Connectivity generation

• Body-fitted mesh + Cartesian mesh + Overlap regions
• Body-fitted + Cartesian points - 26 neighbours based on (i, j, k) index
• ✁

• Problems with FAME mesh

– Not tightly coupled - lead to insufficient and poor stencils – Special, geometry-resolving grids absent – Blending is not good – Insufficient/incorrect overlap information

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Examples of bad connectivity

• ✁

• ✁

X Y

• ✁

X Y

• ✁

X Y

How to quantify bad connectivity ?

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Tests for connectivity

• Minimum number of nodes - over-determined system
• Non-degenerate, determinant = 0 - all points must not lie on a line (2-D) or disc (3-D)
• Least squares - well conditioned

     ∆x1 ∆y1 ∆z1 ∆x2 ∆y2 ∆z2 . . . . . . . . . ∆xN ∆yN ∆zN     

• X(C),

N×3

  ∂xF ∂yF ∂zF   =      F1 − Fo F2 − Fo . . . FN − Fo      = ⇒ X⊤X

A

∇F = X⊤∆F

• Singular value decomposition

X = U · diag(sx, sy, sz) · V ⊤ U = N × 3, column-orthonormal and V = 3 × 3, orthonormal

• SVD solution

∇F = V · diag(1/sx, 1/sy, 1/sz) · U ⊤ · ∆F

• s = 0 or s ≈ 0 =

⇒ ill-conditioned

• Example: If sz = 0 then all points lie on xy-plane =

⇒ cannot determine z-derivative

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• Example: If {∆x ≤ 0} is ill-conditioned

dUo dt + ∂ ∂xG X+

• ∆x≤0

+ ∂ ∂xG X−

• ∆x≥0

+ ∂ ∂yGY +

• ∆y≤0

+ ∂ ∂yGY −

• ∆y≥0

+ ∂ ∂zG Z+

• ∆z≤0

+ ∂ ∂zG Z−

• ∆z≥0

= 0

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• Example: If {∆x ≤ 0} is ill-conditioned

dUo dt + ∂ ∂xG X+

• +

∂ ∂xG X−

• ∆x≥0

+ ∂ ∂yGY +

• ∆y≤0

+ ∂ ∂yGY −

• ∆y≥0

+ ∂ ∂zG Z+

• ∆z≤0

+ ∂ ∂zG Z−

• ∆z≥0

= 0

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Trailing edge of fuselage

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Connectivity near trailing edge of fuselage

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Wing-fuselage intersection

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Connectivity near wing-fuselage intersection ∂ ∂xG X+ From preprocessor After enhancement

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Polynomial accuracy test

• Good stencil ⇐

⇒ better accuracy

• p(x, y, z) = x2 + y2 + z2 + xy + yz + zx + x + y + z
• First order accurate derivative

error in derivative = |(∂xp)computed − (∂xp)exact| = Kh, K = O(1)

• Polynomial test

– If K > TOL - enhance connectivity – If still K > TOL - switch to full stencil Axis x y z Split Stencil C1, C2 C3, C4 C5, C6 Derivative ∂xp ∂yp ∂zp h max |∆xj| max |∆yj| max |∆zj|

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Polynomial test (initial) Deriv Err Err/h Full stencil: XDERV = 1.0275 2.7465e-02 0.31 YDERV = 1.0127 1.2731e-02 0.14 ZDERV = 1.2493 2.4928e-01 1.06<-- Half stencil: C1 = 0.9357 6.4255e-02 0.96 C2 = 1.0998 9.9780e-02 1.13 C3 = 0.8826 1.1739e-01 1.27 C4 = 1.1021 1.0206e-01 1.27 C5 = -0.3021 1.3021e+00 152.23<-- C6 = 1.2618 2.6184e-01 1.12

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Polynomial test (enhanced) Deriv Err Err/h Full stencil: XDERV = 1.0564 5.6380e-02 0.43 YDERV = 0.9374 6.2562e-02 0.44 ZDERV = 1.1671 1.6705e-01 0.71<-- Half stencil: C1 = 0.9051 9.4934e-02 1.41 C2 = 1.1475 1.4747e-01 1.12 C3 = 0.8353 1.6469e-01 1.15 C4 = 1.1073 1.0729e-01 1.34 C5 = 0.8227 1.7729e-01 1.49<-- C6 = 1.2618 2.6184e-01 1.12

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Polynomial test (initial) Deriv Err Err/h Full stencil: XDERV = 1.0051 5.1409e-03 0.09 YDERV = 1.0224 2.2408e-02 0.25 ZDERV = 1.0661 6.6143e-02 2.12<-- Half stencil: C1 = 0.9459 5.4129e-02 0.96 C2 = 1.0499 4.9920e-02 1.60 C3 = 0.9682 3.1757e-02 1.02 C4 = 1.0569 5.6895e-02 0.62 C5 = 0.6655 3.3452e-01 58.23<-- C6 = 1.0709 7.0938e-02 2.27

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Polynomial test (enhanced) Deriv Err Err/h Full stencil: XDERV = 1.0404 4.0352e-02 0.64 YDERV = 1.0255 2.5517e-02 0.28 ZDERV = 1.0480 4.8013e-02 1.54<-- Half stencil: C1 = 0.9511 4.8881e-02 0.78 C2 = 1.0712 7.1240e-02 1.14 C3 = 0.9609 3.9135e-02 0.49 C4 = 1.0711 7.1128e-02 0.78 C5 = 0.7190 2.8101e-01 26.07<-- C6 = 1.0609 6.0853e-02 1.95

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Results with LSKUM and FAME

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M165 configuration

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Cartesian mesh for M165 configuration

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Body-fitted mesh for M165 configuration

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M165 configuration

• Five body-fitted grids - 2 canards, 2 wings, 1 fuselage
• Spherical far-field grid

Fuselage 4 × 149 × 64 Wing 8 × 33 × 64 Canard 8 × 21 × 32 Outer Boundary 4 × 33 × 16

• Background Cartesian grid
• Total number of points = 429358
• Defective split stencils according to SVD test = 284
• Defective split stencils according to polynomial test = 4284
• Split stencils switched to full = 900
• M∞ = 0.789, α = 4.36o

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M165 - surface point distributions

X Y Z Y X Z

Top view Side view

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M165 - pressure coefficient on wing At 74%, 86%

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1

• Cp

x/c 74% wing span LSKUM FAME WT

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1

• Cp

x/c 86% wing span LSKUM FAME WT

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Cp along symmetry plane of fuselage

• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4

• 1
• 0.5

0.5 1 1.5 2 2.5

• Cp

x Cp on vertical symmetry plane of fuselage LSKUM FAME

Pressure 1.056 1.002 0.947 0.892 0.837 0.782 0.727 0.672 0.617 0.562 0.507 0.452 0.397 0.343 0.288 0.233 0.178 0.123 0.068 0.013

• 0.042
• 0.097
• 0.152
• 0.207
• 0.262
• 0.316
• 0.371
• 0.426
• 0.481
• 0.536

Mach 0.829 0.818 0.807 0.796 0.785 0.774 0.764 0.753 0.742 0.731 0.720 0.710 0.699 0.688 0.677 0.666 0.656 0.645 0.634 0.623 0.612 0.602 0.591 0.580 0.569 0.558 0.548 0.537 0.526 0.515

Pressure Mach no

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Kinetic Meshless Method

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Why a new scheme ?

• LSKUM - upwinding through stencil sub-division
• One-sided stencils give correct domain of dependence
• At each point we need four stencils in 2-D (six in 3-D)

C1 = {Pj ∈ Co | xj ≤ xo} − → ∂ ∂xG X+ C2 = {Pj ∈ Co | xj ≥ xo} − → ∂ ∂xG X− C3 = {Pj ∈ Co | yj ≤ yo} − → ∂ ∂yGY + C4 = {Pj ∈ Co | yj ≥ yo} − → ∂ ∂yGY −

• Each sub-stencil must satisfy some conditions
• Leads to a small increase in stencil support
• Problem of insufficient neighbours near boundaries

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Kinetic Meshless Method

• Use least squares, kinetic theory and full stencil
• Upwinding through introduction of a mid-point state (Morinishi, Balakrishnan)
• ✂✁

P

• Pj

s j nj

j/2

F

• Kinetic upwind approximation

Fj/2 = Fo if v · ˆ nj ≥ 0 Fj if v · ˆ nj ≤ 0

• Use mid-point states for least squares approximation

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• Semi-discrete scheme

dUo dt +

• [aj(G

Xj/2 − G Xo) + bj(GYj/2 − GYo)] = 0

• (aj, bj) - geometric coefficients - compute once and store
• Edge-based updating possible - speed up of 2
• On Cartesian points, KMM reduces to finite volume method
• Nearly positive scheme - maximum principle - good stability properties

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Rotational invariance

• Ideally, numerical solution should be invariant to coordinate system
• LSKUM is not invariant - split stencils change with rotation of axes

x y x’ y’ S S’

• Rotational invariance: S

R

− →S′ ρ′ = ρ, E′ = E, ux′ uy′

• = R

ux uy

• KMM approximation

div( vF)o =

• j

[(vx, vy) · (aj, bj)](Fj/2 − Fo)

• Invariant algorithms - a pre-requisite for multi-dimensional schemes

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Higher order scheme

• Least squares formula has linear consistency
• Upwind definition of Fj/2 destroys linear consistency - only zeroth order consistency is
• btained
• Linear consistency - via reconstruction (like finite volume schemes)
• Define left and right states at each mid-point using linear reconstruction along the ray

V +

j/2 = Vo + 1

2∆ rj · ∇Vo and V −

j/2 = Vj − 1

2∆ rj · ∇Vj

• ✂✁

V P P mid−point

j

• +

V−

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Numerical order of accuracy ∂u ∂t + y∂u ∂x − x∂u ∂y = 0

u=0 u=u x y (1,0) (0,1)

• (0,0)
• utflow boundary

u(x, y) = uo(

• x2 + y2)

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Uniform and random points, and solution

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Numerical order of accuracy

• 2.8
• 2.6
• 2.4
• 2.2
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 2.1
• 2
• 1.9
• 1.8
• 1.7
• 1.6
• 1.5

log(Error) log(h) L1 L2 Linf Curve fit

• 2.6
• 2.4
• 2.2
• 2
• 1.8
• 1.6
• 1.4
• 1.2
• 1
• 2
• 1.9
• 1.8
• 1.7
• 1.6
• 1.5

log(Error) log(h) L1 L2 Linf Curve fit

Point distribution L1 L2 L∞ Uniform 2.27 2.21 1.97 Random 2.19 2.12 1.90

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Results using KMM

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Flow over Williams airfoil Free-stream Mach number = 0.15 Angle of attack = 0 Number of points = 6415 Points on main airfoil = 234 Points on flap = 117 n KMM q-LSKUM 3 0.19

• 4

4.86 0.02 5 6.84 0.12 6 80.45 71.36 7 6.50 27.79 8 0.16 0.69 9

• 0.02

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William’s airfoil Pressure coefficient

• 2

2 4 6 8 10 0.2 0.4 0.6 0.8 1

• Cp

x MAIN AIRFOIL q-LSKUM KMM EXACT

• 1

1 2 3 4 5 6 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

• Cp

x FLAP q-LSKUM KMM EXACT

S = pρ−γ (pρ−γ)∞ − 1 Scheme Cl Cd Smin Smax q-LSKUM 3.0927 0.0197 −1.535 × 10−3 1.031 × 10−2 KMM 3.7608 0.0069 −4.99 × 10−4 7.246 × 10−3 Potential 3.736

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Flow over cylinder M∞ = 0.38 and α = 0 No of points = 4111 Points on cylinder = 250 n 4 5 6 7 8 9 KMM 6.67 5.50 83.12 4.72

• LSKUM
• 0.19

82.56 16.78 0.44 0.02 Pressure Mach

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Subsonic flow over cylinder

q-LSKUM entropy, min = -0.00634187, max = 0.0222672 KMM entropy, min = -0.00138786, max = 0.000165191

q-LSKUM KMM Scheme Cl Cd Smin Smax q-LSKUM 0.0237 0.0324

• 0.00634

0.022267 KMM 0.0006 0.0012

• 0.00138

0.000165

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Suddhoo-Hall airfoil

• Four element airfoil
• Number of points = 14091
• On airfoils = 229, 196, 217, 157
• Mach = 0.2 and α = 0

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Suddhoo-Hall airfoil 1 2 3 4 KMM 0.5387 4.8095 2.0925 0.7065 Exact 0.5215 4.7157 2.0794 0.7216 % Error 3.3 1.9 0.6

• 2.0

Circulation around different elements

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NACA-0012 airfoil M∞ = 0.85, α = 1o Adapted points, 3777 Mach contours

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Scramjet Intake - initial solution Inlet Mach number = 5

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Scramjet Intake - adapted solution Inlet Mach number = 5

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Scramjet Intake

Shock Wave angle Velocity

1 2 3 4 Computed 16.44 21.80 18.12 20.32 Exact 16.43 21.72 18.23 20.18 Computed and exact wave angles for scramjet.

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Cartesian Points NACA-0012 - coarse M∞ = 1.2, α = 0o Points Mach number (Mohan Varma)

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Cartesian Points NACA-0012 - adapted M∞ = 1.2, α = 0o Points Mach number (Mohan Varma)

ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit

Credits Ashok Kumar, Sushma Das, Beena O S, Manjula C

Thank You