Shock wave boundary layer interaction in intakes at incidence - PowerPoint PPT Presentation

Introduction Modeling Test case Intake case References Shock wave boundary layer interaction in intakes at incidence Giacomo Castiglioni, Francesco Montomoli, Joaquim Peir´ o, Spencer J. Sherwin Department of Aeronautics Imperial College London g.castiglioni@imperial.ac.uk June 10-12, 2019 Nektar++ Workshop 1/46

Introduction Modeling Test case Intake case References Overview 1 Introduction 2 Modeling 3 Test case 4 Intake case References 5 2/46

Introduction Modeling Test case Intake case References Introduction 3/46

Introduction Modeling Test case Intake case References Motivation Shock wave boundary layer interaction (SWBLI) is a phenomena encountered in many industrial devices (external aero, engine intakes, cascades, nozzles, etc.) and plays a critical role due to its importance for both efficiency and structural integrity, often being the limiting factor to the design envelope. Goal Simulate a simplified, but representative, intake geometry with a high-order, unstructured compressible solver (Nektar++). [KT18] 4/46

Introduction Modeling Test case Intake case References Modeling 5/46

Introduction Modeling Test case Intake case References Nektar++ High fidelity, scale resolving simulations (DNS, uDNS) Framework Spectral h/p element method Unstructured Compressible / Incompressible Target High-Reynolds numbers Complex geometries Transient phenomena www.nektar.info 6/46

Introduction Modeling Test case Intake case References Discontinuous Spectral Element Methods (DSEM) Geometrical flexibility Good dissipation/dispersion properties ‘Natural’ framework for iLES/uDNS Compact schemes 7/46

Introduction Modeling Test case Intake case References Compressible Navier-Stokes equations ∂ q ∂ t + ∇ · ( f ( q ) − g ( q )) = 0 , (1)       ρ ρ u j 0  ,  ,  , q = ρ u i f ( q ) j = ρ u i u j + p δ ij g ( q ) j = τ ij    E ( E + p ) u j u i τ ij − o j (2) p = ρ RT , e = C v T , h = C p T , (3) � ∂ u i + ∂ u j − λ∂ u i � + ζ ∂ u i τ ij = µ δ ij δ ij , (4) ∂ x j ∂ x i ∂ x i ∂ x i o i = − κ∂ T . (5) ∂ x i 3 , ζ = 0, k = C p µ (with λ = 2 Pr ) 8/46

Introduction Modeling Test case Intake case References Laplacian viscosity The RHS of the Navier-Stokes equations is augmented by Laplacian viscosity [PP06] ∇ · ( ε ∇ q ) , (6) for consistency ε ∼ h / p , and from physical considerations ε ∼ λ max = | u | + c [BD10] h ε = ε 0 p λ max S , (7) ε 0 = O (1), S sensor. 9/46

Introduction Modeling Test case Intake case References Physical viscosity Based on a shock sensor artificial shear viscosity and thermal conductivity are added to the physical ones, i.e. µ = µ ph + µ av , ζ = ζ ph + ζ av , κ = κ ph + κ av , (8) Minimal physical viscosity model µ av = µ 0 ρ h p λ max S , (9) C p κ av = µ av Pr , (10) ζ av = 0 . (11) ε 0 = O (1) 10/46

Introduction Modeling Test case Intake case References Resolution based sensor As Shock sensor, a modal resolution-based indicator can be used � � q − ˜ q , q − ˜ q � � s e = log 10 , (12) � q , q � where �· , ·� represents a L 2 inner product, q and ˜ q are the full and truncated expansions of a state variable N ( P ) N ( P − 1) � � q ( x ) = q i φ i , ˆ q ( x ) = ˜ q i φ i , ˆ (13) i =1 i =1 constant element-wise sensor  0 , s e ≤ s k − k ,   � � 1 + sin π ( s e − s k ) 1 S ε = , | s e − s k | ≤ k , (14) 2 2 k s e ≥ s k + k ,  1 ,  s k ∼ log 10 ( p 4 ) (from Fourier coefficients decaying as 1 / p 2 ). 11/46

Introduction Modeling Test case Intake case References Vorticity sensor The aim is to avoid excessive dissipation in regions of high vorticity Ducros’ sensor ( ∇ · u ) 2 S ω = ( ∇ · u ) 2 + |∇ × u | 2 + ε, (15) then the applied sensor becomes S = S ε S ω , (16) 12/46

Introduction Modeling Test case Intake case References Smoothing operators Ducros’ sensor should be 0 ≤ S ω ≤ 1 AV should be strictly positive element-wise constant AV might induce oscillations Strategy Average Ducros’ sensor over an element Compute AV Approximate C 0 projection of AV 13/46

Introduction Modeling Test case Intake case References Soft max function e Ka + e Kb � � Smax ( a , b ) = log , (17) K 10 0 10 − 1 10 − 2 10 − 3 x min 10 − 4 10 − 5 10 − 6 10 − 7 10 − 8 10 − 11 10 − 9 10 − 7 10 − 5 10 − 3 10 − 1 x Applied to pressure Allows for the Riemann solver to work through negative pressure oscillations 14/46

Introduction Modeling Test case Intake case References Test case 15/46

Introduction Modeling Test case Intake case References Test case SWBLI studied experimentally and numerically by Degrez et al. [DBW87]. Conditions M = 2 . 15, β = 30 . 8, p 0 = 1 . 07 × 10 4 Pa , T 0 = 293 K , Re x sh = 10 5 , Pr = 0 . 72. 16/46

Introduction Modeling Test case Intake case References Setup The inflow boundary is located at x = 0 . 3 x sh where the analytical compressible boundary layer solution [WC06] is imposed. Rankine-Hugoniot relations are added to impose the shock. 17/46

Introduction Modeling Test case Intake case References Mesh and Mach number field 120 × 40 elements p = 4 18/46

Introduction Modeling Test case Intake case References Cases considered no AV AV AV + Ducros AV + C 0 AV + Ducros + C 0 19/46

Introduction Modeling Test case Intake case References Pressure and skin friction distribution (no AV) 1 . 6 0 . 004 1 . 5 0 . 003 1 . 4 p/p min 0 . 002 1 . 3 C f 1 . 2 0 . 001 1 . 1 0 . 000 1 . 0 − 0 . 001 0 . 9 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 x/x sh x/x sh Blue line DIRK-2; Red line SSP RK-2; Circles [DBW87]; triangles [BRCD06]; dotted line is empirical solution by [Eck55] for C f or the Rankine-Hugoniot relations for p . 20/46

Introduction Modeling Test case Intake case References Density at horizontal line (no AV) y/x sh = 0 . 1 1 . 4 1 . 3 ρ/ρ ∞ 1 . 2 1 . 1 1 . 0 0 . 25 0 . 50 0 . 75 1 . 00 1 . 25 1 . 50 1 . 75 2 . 00 x/x sh Blue line DIRK2; Red line SSP RK2; Simulation is stable Non-physical oscillations 21/46

Introduction Modeling Test case Intake case References AV case 1 . 6 0 . 004 1 . 5 0 . 003 1 . 4 p/p min 0 . 002 1 . 3 C f 1 . 2 0 . 001 1 . 1 0 . 000 1 . 0 − 0 . 001 0 . 9 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 y/x sh = 0 . 1 x/x sh x/x sh 1 . 3 Black line no AV SSP RK2; Blue line DIRK2; Red line SSP RK2 ρ/ρ ∞ 1 . 2 Non-physical oscillations reduced Challenging to add dissipation only to the 1 . 1 shock! s k = 0 . 25, k = 0 . 75 1 . 0 0 . 8 1 . 0 1 . 2 x/x sh 22/46

Introduction Modeling Test case Intake case References Effects of anti-vorticity sensor and smoothing The Ducros’ sensor lowers the artificial viscosity in regions that have low resolution and high vorticity (small effect in laminar case) 23/46

Introduction Modeling Test case Intake case References Ducros case (AV+Ducros) 1 . 6 0 . 004 1 . 5 0 . 003 1 . 4 p/p min 0 . 002 1 . 3 C f 1 . 2 0 . 001 1 . 1 0 . 000 1 . 0 − 0 . 001 0 . 9 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 y/x sh = 0 . 1 x/x sh x/x sh Black line no AV SSP RK2 1 . 3 Blue line DIRK2; Red line SSP RK2 ρ/ρ ∞ 1 . 2 Non-physical oscillations are almost gone 1 . 1 Still difficult to find stable AV parameters 1 . 0 s k = 0 . 00, k = 0 . 75 0 . 8 1 . 0 1 . 2 x/x sh 24/46

Introduction Modeling Test case Intake case References Smoothing only case (AV+C0) 1 . 6 0 . 004 1 . 5 0 . 003 1 . 4 p/p min 0 . 002 1 . 3 C f 1 . 2 0 . 001 1 . 1 0 . 000 1 . 0 − 0 . 001 0 . 9 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 y/x sh = 0 . 1 x/x sh x/x sh 1 . 3 Black line no AV SSP RK2 ρ/ρ ∞ 1 . 2 Blue line DIRK2; Red line SSP RK2 Shock tube param for AV 1 . 1 Non-physical oscillations are gone 1 . 0 0 . 8 1 . 0 1 . 2 x/x sh 25/46

Introduction Modeling Test case Intake case References Ducros and smoothing case (AV+Ducros+C0) 1 . 6 0 . 004 1 . 5 0 . 003 1 . 4 p/p min 0 . 002 1 . 3 C f 1 . 2 0 . 001 1 . 1 0 . 000 1 . 0 − 0 . 001 0 . 9 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 y/x sh = 0 . 1 x/x sh x/x sh Black line no AV SSP RK2 1 . 3 Blue line DIRK2; Red line SSP RK2 ρ/ρ ∞ 1 . 2 Non-physical oscillations are gone 1 . 1 Shock tube param for AV Ducros’ sensor has little effect 1 . 0 (laminar flowfield) 0 . 8 1 . 0 1 . 2 x/x sh 26/46

Introduction Modeling Test case Intake case References Flat plate summary Good quantitative agreement for 2D laminar SWBLI C 0 smoothing increases robustness and decrease influence of AV parameters C 0 smoothing allows for a ‘sharper’ AV Ducros’ sensor helps less than C 0 smoothing (laminar flowfield) 27/46

Introduction Modeling Test case Intake case References Intake case 28/46

Introduction Modeling Test case Intake case References Inviscid case, Mach number distribution Mach = 0 . 435, α = 23 . 15. Total pressure is imposed at the inlet, static pressure at the outlet. 29/46