Verification and Validation of Pseudospectral Shock Fitted Simulations of Supersonic Flow over a Blunt Body Gregory P. Brooks Air Force Research Laboratory, WPAFB, Ohio Joseph M. Powers University of Notre Dame, Notre Dame, Indiana 42nd AIAA Aerospace Sciences Meeting 6 January 2004, Reno, Nevada AIAA-2004-0655 Support: U.S. Air Force Palace Knight Program

Motivation

• Develop verified and validated high accuracy flow solver for Euler equations in space and time – verification: solving the equations “right” – validation: solving the right equations

• ultimate use for fundamental shock stability questions for inert and reactive flows, detonation shock dynamics, shape optimization

Review: Blunt Body Solutions

• Lin and Rubinov, J. Math. Phys., 1948 • Van Dyke, J. Aero/Space Sci., 1958 • Evans and Harlow, J. Aero. Sci., 1958 • Moretti and Abbett, AIAA J., 1966 • Kopriva, Zang, and Hussaini, AIAA J., 1991 • Kopriva, CMAME, 1999 • Brooks and Powers, J. Comp. Phys., 2004 (to appear)

Model: Euler Equations

• two-dimensional • axisymmetric • inviscid • calorically perfect ideal gas

Model: Euler Equations



∂u ∂w u ∂ρ ∂ρ ∂ρ +u +w +ρ + + ∂t ∂r ∂z ∂r ∂z r



=0

∂u ∂u ∂u 1 ∂p +u +w + =0 ∂t ∂r ∂z ρ ∂r ∂w ∂w 1 ∂p ∂w +u +w + =0 ∂t ∂r ∂z ρ ∂z   ∂p ∂p ∂u ∂w u ∂p + u + w + γp + + =0 ∂t ∂r ∂z ∂r ∂z r

Model: Secondary Equations

∂u ∂w ωθ = − ∂z ∂r ωθ dρ 1 dωθ = + 2 dt ρ dt ρ



∂ρ ∂p ∂ρ ∂p − ∂z ∂r ∂r ∂z 

p 1 p , s = ln γ T = γ − 1ρ ρ



,



u + ωθ r

ds =0 dt


γ p 1 2 2 Ho = + u + w = constant γ − 1ρ 2

Flow Geometry and Boundary Conditions r 1.2

• body: zero mass flux

ξ

1 Shock

h(ξ,τ)

• shock: RH jump

0.8 0.6 v∞

• center: homeoentropic

Body (R=Z b)

0.4

• outflow: supersonic

0.2

η

z −0.4

−0.2

0.2

0.4

0.6

0.8

1

Flow Geometry in Transformed Space η Shock 1

• (r, z, t) → (ξ, η, τ )

0.8 0.6 Centerline

• unsteady

Outflow

0.4

• shock-fitted to avoid low

0.2

0

.

0.2

.

.

0.4 0.6 Body

.

0.8

1

ξ

first order accuracy of shock capturing

Outline: Pseudospectral Solution Procedure

• Define collocation points in computational space. • Approximate all continuous functions and their spatial derivatives with Lagrange interpolating polynomials,

which have global support for high spatial accuracy. spatial discretization

algebra

• PDEs −−−−−−−−−−−−→ DAEs −−−−→ ODEs. • Cast ODEs as

dx dt

= q(x).

• Solve ODEs using high accuracy solver LSODA.

Taylor-Maccoll: Flow over a Sharp-Nose Cone

• Similarity solution

r

available for flow over

ξ

1.2

a sharp cone

1

M∞

0.8

Shock Body

0.6 η

flow field

0.4 0.2

−0.4 −0.2

• Non-trivial post-shock

ro 0.2

0.4

0.6

0.8

1

z

• Ideal verification benchmark

Verification: Taylor-Maccoll Time-Relaxation 0

• M∞ = 3.5

−5

10

• 5 × 17 grid • t → ∞, error → 10−12

−10

10

∞

L [Ω] residual in ρ

10

steady state error −15

10

2

4

τ

6

8

10

Verification: Taylor-Maccoll Spatial Resolution

• spectral convergence

0

L∞[Ω] error in ρ

10

• roundoff error realized

−5

10

at coarse resolution,

5 × 17

−10

10

−15

10

0

10

1

2

10 10 number of nodes in η direction

3

10

• run time ∼ 102 s; 800 M Hz machine

Blunt Body Flow: Mach Number Field 1.5

1.8

1.8

1

1.6

• M∞ = 3.5

1.6

r

1.4

1.4

• 17 × 9 grid

1.2

0.5

√ •R= Z

1

• transonic flow field

0.8

0.6

predicted

0 −0.2

0.

2

4 0.

0

0.2

0.4 z

0.6

0.8

• qualitatively correct • not a verification

Blunt Body Flow: Pressure Field

5.5

1.5

6.5

1

7.5

r

8.5

11

.5

16

15

13.5

0 −0.2

0

0.2

0.4 z

• qualitatively correct • not a verification

10

0.5

• high pressure at nose

0.6

0.8

Blunt Body Flow: Vorticity Field 1.5

• Helmholtz Theorem: dρ dt , ∇p × ∇ρ, shock

-1 -1.5 1 -2 r

curvature, flow

dω

-3

divergence induce dtθ

-4

0.5

• intuition difficult

-3 -2 -1 0 -0.2

0

0.2

0.4 z

0.6

0.8

1

• not a verification

Verification: Blunt Body Pressure Coefficient

• Cp =

2p(ξ,0,τ )−1 2 γM∞

• Newtonian theory gives

1.8

prediction in high Mach

Pseudospectral prediction Modified Newtonian theory

1.6 1.4

number limit

C

p

1.2 1

• comparison quantitatively

0.8 0.6

excellent

0.4 0.2 0

0.2

0.4

0.6 r

0.8

1

• not global

Verification: Blunt Body Entropy Field 1.5

•

0.3

=

∂s ∂t

• if stable, t→∞

1 0.4 r 0.5

+∇·v =0 ∂s ∂t

→ 0 as

• thus, v · ∇s → 0

0.5

• quantitative difference

0.6

0 -0.2

ds dt

approaches roundoff

0

0.2

0.4

z

0.6

0.8

1

error

Proof: Total Enthalpy is Constant

• Ho ≡

γ p γ−1 ρ

+

1 2

u2 + w


2

(definition)

ds ∂p = ρT + (from Euler equations) • dt ∂t |{z} |{z} =0 →0 • Ho = constant on streamline as t → ∞ o ρ dH dt

• RH shock jump equations admit no change in Ho • If Ho is spatially homogeneous before the shock, it will remain so after the shock; Ho = constant. QED.

Verification: Blunt Body Total Enthalpy

−5

x 10 0.5

1.5

0 1

−0.5

r

−1 −1.5

0.5

−2 0

−2.5 0

0.5 z

1

• Ho : a true constant • deviation from freestream value measures error

• 17 × 9, error ∼ 10−5 • 29 × 15, error ∼ 10−9 • good quantitative verification

• “exact solution” from 65 × 33 grid

−5

10

• spectral convergence

∞

L [Ω] error in ρ

Verification: Blunt Body Grid Convergence

−10

10

• error → 10−12 1

10

2

10 number of nodes

3

10

• best quantitative verification

Validation: Flow over a Sphere 1.4

• Shock shape

1.2

predictions match

1 0.8 r

Billig’s (JSR,

0.6

1967)

0.4 Body surface Pseudospectral prediction Billig

0.2 0 −0.6

−0.4

−0.2

0

0.2 z

0.4

0.6

0.8

1

• Error ∼ 10−2

Unsteady Problem: Acoustic Wave/Shock Interaction

• low-frequency

freestream input

0

10

∆ρ |

∞ z=−1

∆ ρ∞

−2

10

∆h

• low-amplitude,

−4

10 P(fk)

disturbance

−6

10

high-frequency

−8

10

response captured by

−10

10

−12

10

0

20

40 60 reduced frequency (fk)

80

100

high accuracy method

• 33 × 17 grid; run time, 7.5 hrs.

Conclusions

• Pseudospectral method coupled with shock fitting gives solutions with high accuracy and spectral

convergence rates in space for Euler equations.

• Standardized formulation of

dx dt

= q(x) allows use of

integration methods with high accuracy in time.

• Algorithm has been verified to 10−12 . • Predictions have been validated to 10−2 . • Discrepancy between prediction and experiment is not attributable to truncation error.

• Challenge to determine which factor (e.g. neglected physical mechanisms, inaccurate constitutive data,

measurement error, etc.) best explains the remaining discrepancy between prediction and observation.

• Challenge also to exploit verification and validation for first order shock capturing methods, necessary for complex geometries.