Verification and Validation of Pseudospectral Shock Fitted - PowerPoint PPT Presentation

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

Model: Secondary Equations ω θ = ∂u ∂z − ∂w ∂r � ∂ρ � dt + 1 dω θ dt = ω θ dρ ∂p ∂r − ∂ρ ∂p u + ω θ ρ 2 ρ ∂z ∂r ∂z r � p � 1 p ds T = s = ln dt = 0 ρ, , ρ γ γ − 1 ρ + 1 γ p � u 2 + w 2 � H o = = constant γ − 1 2

Flow Geometry and Boundary Conditions r 1.2 • body: zero mass flux ξ Shock 1 h( ξ , τ ) • shock: RH jump 0.8 • center: homeoentropic 0.6 Body (R=Z b ) v ∞ 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 • unsteady Centerline Outflow 0.4 • shock-fitted to avoid low 0.2 first order accuracy of . . . . 0.2 1 0 0.4 0.6 0.8 ξ Body 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 d x 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 ∞ • Non-trivial post-shock 0.8 Shock Body 0.6 flow field η 0.4 • Ideal verification 0.2 r o z benchmark 0.2 0.4 0.6 0.8 1 −0.4 −0.2

Verification: Taylor-Maccoll Time-Relaxation 0 10 • M ∞ = 3 . 5 L ∞ [ Ω ] residual in ρ −5 10 • 5 × 17 grid • t → ∞ , error → 10 − 12 −10 10 steady state error −15 10 2 4 6 8 10 τ

Verification: Taylor-Maccoll Spatial Resolution • spectral convergence 0 10 • roundoff error realized −5 L ∞ [ Ω ] error in ρ 10 at coarse resolution, 5 × 17 −10 10 • run time ∼ 10 2 s ; −15 10 0 1 2 3 10 10 10 10 800 MHz machine η number of nodes in direction

Blunt Body Flow: Mach Number Field 1.5 √ • R = Z 1 . 8 1.8 6 1 1 . • M ∞ = 3 . 5 6 1 . 4 . 1 r 4 1 . • 17 × 9 grid 1.2 0.5 1 • transonic flow field 8 . 0 0.6 predicted 0.4 2 . 0 0 −0.2 0 0.2 0.4 0.6 0.8 • qualitatively correct z • not a verification

Blunt Body Flow: Pressure Field 1.5 • high pressure at nose 5.5 • qualitatively correct 1 6.5 • not a verification 5 r . 7 8.5 0.5 1 0 1 1 . 5 13.5 15 6 1 0 −0.2 0 0.2 0.4 0.6 0.8 z

Blunt Body Flow: Vorticity Field 1.5 • Helmholtz Theorem: -1 dρ dt , ∇ p × ∇ ρ , shock -1.5 1 -2 curvature, flow r divergence induce dω θ -3 dt -4 0.5 • intuition difficult -3 • not a verification -2 -1 0 -0.2 0 0.2 0.4 0.6 0.8 1 z

Verification: Blunt Body Pressure Coefficient • C p = 2 p ( ξ, 0 ,τ ) − 1 γM 2 ∞ • Newtonian theory gives 1.8 prediction in high Mach Pseudospectral prediction Modified Newtonian theory 1.6 number limit 1.4 1.2 C p 1 • comparison quantitatively 0.8 0.6 excellent 0.4 0.2 • not global 0 0.2 0.4 0.6 0.8 1 r

Verification: Blunt Body Entropy Field 1.5 • ds dt = ∂s ∂t + ∇ · v = 0 0.3 • if stable, ∂s ∂t → 0 as 1 0.4 t → ∞ r 0.5 • thus, v · ∇ s → 0 0.5 • quantitative difference 0.6 approaches roundoff 0 -0.2 0 0.2 0.4 0.6 0.8 1 z error

Proof: Total Enthalpy is Constant � u 2 + w 2 � γ p ρ + 1 • H o ≡ (definition) γ − 1 2 dt = ρT ds + ∂p • ρ dH o (from Euler equations) dt ∂t ���� ���� → 0 =0 • H o = constant on streamline as t → ∞ • RH shock jump equations admit no change in H o • If H o is spatially homogeneous before the shock, it will remain so after the shock; H o = constant . QED.

Verification: Blunt Body Total Enthalpy • H o : a true constant −5 x 10 1.5 0.5 • deviation from freestream 0 value measures error 1 −0.5 • 17 × 9 , error ∼ 10 − 5 −1 r −1.5 0.5 • 29 × 15 , error ∼ 10 − 9 −2 • good quantitative −2.5 0 0 0.5 1 z verification

Verification: Blunt Body Grid Convergence • “exact solution” from 65 × 33 grid L ∞ [ Ω ] error in ρ −5 10 • spectral convergence −10 10 • error → 10 − 12 1 2 3 10 10 10 • best quantitative number of nodes verification

Validation: Flow over a Sphere 1.4 • Shock shape 1.2 predictions match 1 0.8 Billig’s ( JSR , r 0.6 1967) 0.4 Body surface Pseudospectral prediction • Error ∼ 10 − 2 0.2 Billig 0 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 z

Unsteady Problem: Acoustic Wave/Shock Interaction • low -frequency freestream input 0 10 disturbance ∆ ρ ∞ | z=−1 ∆ ρ ∞ −2 10 ∆ h • low-amplitude, −4 10 P(f k ) −6 10 high-frequency −8 10 response captured by −10 10 high accuracy method −12 10 0 20 40 60 80 100 reduced frequency (f k ) • 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 d x 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.